Just got back from India a week ago - Mumbai airport will take time to clear, so if you leave on the early morning flight to Goa you will not get more than a couple of hours of sleep at best. The Sheraton by the airport is great and offers a free shuttle to and from the terminals - the web site lists higher rates for rooms with the shuttle but they take you regardless.

Whatever you end up doing, take an extra dose of patience and enjoy Goa.