Well, I'm sure that you've been told already that a baby should never be allowed to fall asleep while feeding- nursing or bottle feeding due to tooth rot possibilities. That said, I have one or two ideas.

First, since he doesn't seem to care whether the bottle is empty or not, put a little bit of water, not milk, in it and let him sleep with that for a little while.

The next step you could try would be to just put the bottle- no liquids, no nipple- just the bottle to hug on.

Be ready for some crying at each step since it has become a comfort item to him and none of us want to get rid of what comforts us (I'd hate to see me without chocolate!). Sit with hiim, but don't rock him to sleep since that will just cause one more problem you'll need to deal with down the road.

The last step is to remove the bottle altogether, although at this point you wouldn't have to. If you do, give him something else to hold onto. One of my boys likes a MomMom-knitted bear and a Tigger. The other one likes an Eeyore, elephant and receiving blanket.
You could try to have him pick out something new because "some other little boy really needs the bottle now" and he gets to find a new thing to sleep with.

As for during the day, congratulations on moving him to a sippy cup. You could also try to get him used to using a regular, uncapped, small, plastic juice cup for water so that he can start learning how to use it at mealtimes. This is something like what I mean, although it being Amazon you get 6 dozen. I got 6 of them for about $3 from Family Dollar or Walmart or something. Since they're only 6 ounces or so, they are small enough for young kids to hold.

Good luck!