You will come to the international arrival area at Mumbai. Unfortunately, the lounges are available only in the international departure area, and there is no way you'll be able to get past security to go to those lounges even with a lounge membership (i.e. priority pass). Security is very tight in Mumbai.
The best you can do is go straight to the domestic terminal. It is like most airports, and you can usually find a couple seats together (without armrests inbetween) where you can lie down and sleep.
For your domestic flight, security will open about 3 hours before, so when they do, go through security to the gate. this gives you 4-5 hours at Mumbai before having to go through security. With the bus to transfer to the domestic terminal (which takes about 30 minutes), and the recheck in process, you're looking at much less time. It will go by pretty quickly.
There are a good number of food shops as well. Eat !! Dont eat "chutneys" or things that are not cooked however, because as a foreigner, you might get sick (water is different in india than in the US. The same thing actually works the other way too -- indians coming to the US generally cant have US tap water without getting sick).