I answered a similar question to this: How Do I Get More Guns?. Noticed that your question was posted a few hours before that though.
This is the new "Arms Deal" update which went live on 14/08/2013.
Valve released a really good FAQ on it, which can be found here.
The bits of interest for your particular question, "How do I get item drops?":
- Timed weapon drops, just by playing on official and community servers (VAC enabled) in any game mode.
- Opening dropped weapon cases with a weapon case key.
- Trade weapons with other players through Steam's Trading interface.
- Purchase them from other players through the Steam Marketplace.
Regarding the first point, these are obtained for play time over the course of a week. As your play time increases, the drop rate will slow down. This is reset when the week rolls over. Not sure the start day or time of a week though.
I'm not 100% sure where you will get the weapon case keys. The below quote suggests that you buy these:
This update will also actively support the CS:GO competitive community with the introduction of the eSports Weapon Case. A portion of the proceeds from sales of eSports Case Keys will go towards prize purses at competitive events.
Hope that helps.
As missions are quite new, I can only tell from my experience:
How missions work in general
Every mission explains itself in its short description. Most of them are "win a complete match on this specific map" or "get a number of kills in this specific game mode". After you completed this mission, you get rewarded with a weapon drop.
A little bit more in detail
You do have 2 missions at start. Every mission is explained quite simple and in order to achieve it / get your reward you need to finish the map you would complete the mission - so don't leave to early.
For example: "Get 6 kills in game mode Destroy." This means you have to look for a game in this game mode. You will join a server and you likely will kill 6 people. After you have killed those guys, be sure you stay until this map is finished.
In the end screen score board, you will receive your mission reward as a weapon drop. (Since now I only got trash drops, I don't know if they cover up uncommon or rare items, too. All items seem to be out of the new added items, no old ones.)
How to get new/more missions
For this you only have to play one supported game in the finding a game screen. This covers up match making, Gungame, Destroy and so on. In the end of the game, where weapons drop, you will now have an additional chance of a mission drop.
So if you need some missions, you may play a few Gungames, as they're finished quite fast and you get the chance for a mission drop after each game.
Best Answer
No, buying and using the access pass after the campaign has supposedly ended, only yields a operation coin. Beyond that there will be no missions or awards to pick up. However, cases from these operation "seasons" can occasionally drop during gameplay, tho it is rare. So, in the end, buying an access pass for an expired operation season is pointless unless you want to collect the coin(s).