Hey Andreas,
You'll find it quite easy to get by in Mumbai on English alone. Anywhere you choose to work, English most likely will be the primary language.
For local transport, if you use Uber, you will again get by with English alone and mind you Uber in India is far cheaper than elsewhere - the most I've paid so far for an Uber ride is a $10 for a 2.5 hrs long journey.
Apart from talking to people on the streets or in public transport, it's pretty much a no problem.