Since you're travelling alone and it's still not decided that your bf will pick you up or not, just book a hotel before you leave. Near the international terminal there are lots of hotels like Leela, Le Meridian, Intercontinental, etc. Nearby you can also opt for Mirador, Kohinoor, Bawa International, etc. If you are arriving late at night, go for a hotel that's near the airport or else if you decide to stay in some other place then choose a hotel that has airport pick up facility. You can stay there for a day or two, before your bf picks you up or you get to choose another cheaper hotel. Don't depend on taxi drivers or agents to look for a hotel on arrival as that way you'll get the worst deal possible.
Since you asked about renting an apartment, it depends on for how much time you'll be staying here and if there's anyone like a relative or somebody of yours who stays here in Mumbai. Unmarried girls can find it pretty tough to get an apartment here in Mumbai to be very frank. You can find one, but it'll take time. Another thing to consider is that, you'll have to pay 1-2 months rent as brokerage and also about 10 months rent as deposit. And depending on the locality that you choose for your stay, a 1 BHK can cost you anywhere between Rs.10,000-20,000 p.m. (If you stay near the airport, i.e. Andheri (E) it'll be atleast Rs.15,000). So, as you can see for short stays renting an apartment can also be pretty costly.
It's not advisable to carry too much of electronic goods as the customs process in India is not very transparent to put it mildly. However, carrying the normal stuff like a laptop, digicam, cell phone, etc. is not a problem.
At the international terminal local taxi and pre-paid taxi of Hertz and Avis are available (no autos in international terminal). If you choose local taxi, make sure that they agree to travel by the meter and not some fare acc. to their wish.
Well that's all that comes to my mind at present. If you want to know something more, feel free to ask.