You have it almost right!
いっぱい can mean a drink like you said, but in this case it means ‘a lot’. Check out definition 5. [Edit: Nevermind, it does mean ‘a drink’ in this case]
のもう is 飲もう, the volitional (I think) form of 飲む (to drink). It basically means ‘let’s drink’ in this case.
Hope that helps
Also, there’s a specific thread for this type of questions, where you’ll probably get faster replies: Short Grammar Questions (Part 2)