Congrats on your certification. The thing about diving deep is that it's not so much different from diving in shallower depths but it does take some time to ensure you are aware of how far away your buddy is, that you feel comfortable with what you are doing, and that you understand the surface is further away. That last item may sound silly but it's not. I've been diving for 2 years, and I've been on several deep dives. They didn't worry me, but they did bother some of my friends a little, and I will give you the best advice I can - don't do any dive you don't feel comfortable doing. Just because your friend wants to do it doesn't mean you have to do it.
Having said that, I've been diving in Cozumel twice, and most of it was drift diving along a wall. Drift dives are different, because the boat isn't tied to something when you get out, and you need to stick with your group so the boat can pick you all up together. I would encourage you to buy a surface marker buoy, whistle, or mirror -- some kind of signalling device. These are not very expensive and you may never need them, but if for some reason you were separated from your group you would have a way to show others where you are. If you buy something, make sure you know how to use it. I would think a signalling device is more important than spare air, which doesn't actually hold much air for a diver who is freaking out and is most useful in getting you to the surface when you are running out but are not panicked. If you have a pressure gauge, and monitor it regularly, you shouldn't run out of air - you should signal your Dive Master and buddy, and start to ascend according to predetermined pressure limits. This is all part of the dive plan that you should establish before you ever get in the water. If your tank has 3000 psi when you start, when you get to 2000 or 1500, what ever you agree on up front, you will slowly ascend, doing your safety stop, and you should be just fine. Remember to inflate your bcd when you hit the surface, and don't take off your fins until you are physically holding on to the boat.
Also, just because you dive deeper doesn't mean you have to exceed the limits. Remember this from class - most of the time you don't go to 60 feet and stay there for 60 minutes on your first dive. You start out with the deepest part of the dive, and then you gradually ascend. Maybe you swim 15 - 20 min at 60 feet, then another bit at 40 feet, and then a while at 20 feet. You can stay within the limits of no decompression diving by this multi level diving, while still getting to see some cool stuff. A lot of diving in Cozumel is along a wall, so you can see interesting things at shallower levels too.
The most important thing is to talk about all of this stuff up front. Maybe call the shop where you get certified and go through these ideas, and possibly read a book or some postings here about Cozumel. Have fun, and dive safely.