Playing short form requires several shifts in technique and approach. My group typically does 2-3 hour sessions. A 4 hour session is a marathon for us.
Drop the Filler
The first thing to do is let go of filler material. Filler material includes setting up adventures that are "clue to clue to clue to oh actually interesting development". This is the default for a lot of play, and it wastes time. Let go of having the players play out haggling for everything, or explain each store they go to for supplies, and so on.
If there are encounters that would take up time but not actually provide interesting choices or hard implications? Cut them out.
There's also a lot of time wasting in putting in situations that make it unclear what the players should even be trying to do. "Where should we go next?" "I guess we keep searching until we find a clue of what to do next?" etc. Videogames used to use mazes to add extra hours of gameplay as a filler device, and this is the same kind of thing.
Let players get to the fun interesting stuff without this obfuscation and play not only goes quicker, it's more fun period.
Scene Framing
Scene framing is the next step. You no longer say, "Ok, where do you want to go next?" as the default question. If you know the general thing the players are doing, you skip up to the next part and get right into it.
For example, you know the players are trying to track down a cult:
"After three days of talking with shady people, listening to wacked
out babbling of street preachers, and having to do some small trading
in hallucinagens, you finally find out where they're having meetings.
At dusk, you find yourselves outside an abandoned church where several people died of plague years
ago..."
Think of how movies or tv shows will give you a montage - so you can skip the legwork and go RIGHT to the interesting stuff. This alone cuts out a LOT of time wasting material.
The Rules you use matter
If you're playing a game where combats take 45 minutes... well, one combat will take up a significant chunk of your time. Understand that many of the older games and traditional games expect you to be playing for 6-8 hours as session, so if they expect 5 combats in a session, and you've got 3 hours to play... it's not going to fit.
If you're playing a game where conflicts of all types, take 5 minutes? Well, then things move much quicker. So pay attention to what rules you're using and what their impact is, so you can plan appropriately.
Getting into character, exploring the world, etc.
I've been playing this way for several years now. My group gets into characters quite well, because the focus of our play is on characters, their issues and personalities, which is because a lot of our game time is putting the characters into crisis points and fun interactions.
I find a lot of people used to playing with lots of time, think the solution to character or world development is simply pouring on more time, when in fact, it's about directed and focused play. I've had several people say, "We've done more in this two hour one shot than I've done in entire campaigns, for years."
That's because I cut out the filler, focus on the characters' choices and actions, and try to give situations that are at turning points. Each scene should matter and have some impact.
Best Answer
For background, I run an online session with people spanning an eight hour time difference. Our scheduling is generally a mess and we vary between 45 minute and 4 hour sessions depending on the day. I've picked up a few tricks to help things move more smoothly.
Plan ahead
A lot of time in-game is spent having either in- or out-of-character discussions about what your next move is going to be. Encourage your players to discuss their options outside of your games (maybe set up a chat or text group between them all) so they walk into each session with a plan ready to go, and only have to spend three minutes explaining it to you instead of thirty five figuring it out.
Get buy-in
Your players should be aware of the time crunch as keenly as you are. Ask for some small sacrifices in their play styles to make the whole game go more smoothly. Maybe your sorcerer needs to study his spells for a couple days until he can make snap decisions on the fly, or your fighter could use flashcards so she knows what attacks she can make.
Basically, encourage your players to find shortcuts in how they play to match the shortcuts you'll be taking. This kind of ties into the plan ahead step above, but is more about players knowing what they can and can't do like the back of their hands.
Be prepared to end early
I've learned to spot good (and sometimes just decent) stopping points and grab them, even if it is a little earlier than our time slot should be ending. Better to end on a high note than get interrupted in the middle of a good scene.
For example, if you're about to infiltrate a hideout and you only have 5mins left in your lunch period, call it quits before they start to enter. No point getting halfway down the first hallway just to have to pause, and you can use the extra five minutes to have them plan their angle of attack for tomorrow instead.
Fudge it
If you know you only have 10 minutes left and they're mid-boss-fight, there's no harm in quietly knocking off 10% of the big baddie's health, or letting him go down if one of your players manages a particularly spectacular roll.
Similarly, you can always have minions escape instead of fighting to the last drop of HP. It's easy to shave a minute or two off your play time just by making the world bend to your PC's wishes. Done carefully and silently enough, they won't even notice that you're fudging things.
Recap early
If your sessions are anything like mine, you'll spend at least five minutes at the beginning recapping what happened last time and fielding questions from your less attentive players. Move this process away from the table and you'll get your five minutes back.
Take a bit of time after each session to write up a couple sentences about what happened, and post it to your group chat. Then your players can read up on it before they get to the table and you can skip the live recap.
Skip the lines; shop online
Some groups may love to roleplay the shopping experience, but if your party is in the take it or leave it camp, this can be a great way to save time. Instead of painstakingly letting each player play through buying / selling / identifying everything each time they hit a town, end the session and take the deals online.
Try to time your shopping sprees to start at the end of your session, then tell everyone to text you their to-do lists. You can work through them one-on-one and get everyone squared away before the next session.
Now this requires two things; they need to trust you'll do right by them, and because of that sometimes you'll need to just say yes. If they say they'd barter for the potion, give them a (reasonable) discount. If they're trying to find a specific type of mirror, agree that they've found it (unless you have really good reason not to) and hash out the price.
This does mean that you can't hide any plot points in their shopping adventure, but it can be a small price to pay for the time you win back.