How to guide Dampe around

While it might be a little underwhelming to only be able to walk around Clock Town for the first cycle, these are some of the routes you could pursue in the meantime:

Collecting rupees Rupees are the main currency of Termina so you will need a few for the rest of the game. You can choose to grind some rupees right now while you're stuck in Clock Town. Don't forget to hand over rupees to the bank, at 200 rupees in the bank you'll get an adult's wallet. The easiest way to grind I found was smashing the grass in North Clock Town over and over by leaving and entering the area.

Deku Scrub Playground The Deku Scrub Playground is a way to make a few rupees and a heart piece. Since you have to go back 3 times in one cycle to get the heart piece, you might as well do it now you're locked in.

Buy a Bomb Bag

Post Office (First Day) The postman might be able to give you a heart piece on the first day between 3PM and 12AM, that's if you're able to beat his minigame.

??? If you collect the first Title Deeds by exchanging the Moon's Tear, it's possible to get another heart piece at the Stock Pot Inn, every day from midnight till morning.

Note: I haven't gotten this far in 3D yet, so this is based on the original N64 version.

There are really three ways to speed up at this point in the game.

1. Roll as often as you can. Which can be tricky as there are a lot of traps around this area.
2. Get the Bunny Hood first.
3. Play the Inverted Song of Time first. You can learn this song from the Scarecrow in the Observatory basement.

To get the Bunny Hood in the 3DS version:

1. Go to the Laundry Pool during the first or second night and listen to the organ-grinder guy. He will give you the Bremen Mask.
2. Go to Romani Ranch on the third day (it's blocked until then). Specifically, go to the Cucco (chicken) Coop.
3. Talk to the guy there.
4. Use the Bremen Mask to start marching around.
5. March near the 10 Cucco chicks and they should follow you.
6. Once all of them are following you, they should all turn into adults.
7. Talk to the guy to get the Bunny Hood.

Edit:

Having done this now, the Bunny Hood does make this a lot easier in the parts where you don't need the Deku mask to proceed.

However, when he goes around corners quickly, it's still possible to lose track of him.

You have to use your ears in this instance. It will always play his helicopter umbrella sound from whichever direction he turned at that corner. As long as you're within a reasonable distance of him, you should be able to get through any doors before they close.