Check out Pranaam service offered through Mumbai Airport Agency.
(Information is available on Airport website)
There are few reviews here somewhere and on TA
Depending on where you have to be in Mumbai, there are many upscale hotels in Mumbai.
Taj at Gateway of India is iconic but it is near southern tip.
JW at Juhu is famous (even showcased in Marriott Documentary on CNBC)
Sofitel in Bandra-Kurla-Complex, Four Seasons in Worli, etc.
Mumbai traffic is bad. So better stay near where you need to be.
Your hotel or host can arrange for meet-n-greet from outside arrival to hotel. (Newer Mercedez for 1 pax, Huge SUV for more). This is very commonly offered service.
Pranaam will meet you at the gate and guide you through immigration and customs. This service is not that widely used as it is considered unnecessary expense by most travelers including western businessmen on company dime. (English is spoken/understood widely in cities like Mumbai)