Ten USD or equivalent is not an appropriate tip in Mumbai for a driver taking you to the old town... a prepaid taxi charges the equivalent of about 8-9USD for that whole journey. You should really tailor your tipping to the culture and the economy... an equivalent of a couple of dollars is generally deemed quite generous for that trip.
To get local currency, there are forex desks plus ATMs in the arrivals hall after customs and before the exit to the outdoor meeting area. Your driver will be outside at the railings with all the others, so you have time to change or withdraw some INR, there are also a couple of shops and cafes in that arrival hall so you'll be able to break larger notes before you even meet the driver.