Your GOI (Goa) - BOM flight will land at the domestic airport and international flight will depart from the international airport. Even though they share the same run way they are considered seperate airports and are about 5 kms apart.

Bombay traffic isn't too bad between 11:00 pm and 5:30 am so it shouldn't take you more than 20 - 30 minutes to get from the domestic to international terminal. I haven't taken a prepaid taxi from the airport in a while but I'm guessing it shouldn't cost you more than Rs. 200.

You can also check out the shuttle buses that run between the domestic and international terminal but I think a taxi would be quicker and easier at the hour you're landing.

The international airport has plenty of restaurants, coffee shops, stores, etc. I think their might be a lounge or two that offers a day pass.

The other option will be to take a taxi to a hotel located near the international airport and having a snack there and just hanging out. Several of the hotels have a free shuttle to the international airport.