Hi Milo,
I usually stay at The Leela hotel. It is quite near the airport and they provides pick up from the airport to hotel using hotel limousine at free of charge. You can check with the hotel that you are going stay if they provide a free airport pick up and drop off.
For first timer, I will suggest you have your meals at the hotel. In case you are not used with the food there, it's too spicy for me. I find the garlic/butter naan tasted much better than prata. Some hotels do have western food, so you can have more choices.
It's always save to drink from bottled water, local brands are fine, such as Kingfisher or Bisleri. It costs around 20-30Rs. Just ensure the seal is not broken. You don't really have to go for Evian or other foreign brands.
Just like any other countries, pickpockets are everywhere, be careful of your bags and belongings.
Tipping in hotel is not a must but they would really appreciate if you do so, can be ranging from 50 - 100Rs.
Mumbai airport is quite modern now, they have Coffee Bean, Pizza Hut, and KFC. You can take your meals there.
You may wish to request your company over there to arrange for a transport. There are many transport companies available in Mumbai. They are usually 6 seaters family car (similar to Toyota Picnic but more humble).
Hygiene and cleanliness is quite bad. Could be quite humid and dusty, you might pack lip balm and hand cream.
That's all from me, enjoy your first trip to India and wish you a pleasant journey!