Journey from India (Bombay/Mumbai) to USA on ship

I can't speak to the time taken but you can probably start with Freighter Expeditions as a starting point.

As far as I can tell there is only one route that enters a port in India and it originates in Malaysian Port Tanjung Pelepas. The entire journey takes 21 days but I don't think that India -> Malaysia takes that long but you will need to call the provider to find out. One thing to note that you will need to get to Singapore somehow, which you can do on a ship in many different ways and you can look at Maris Freighter Cruises for various journeys that stop at Tanjung Pelepas (there are a few). And then change the ship.

What I read about it suggests that freighter travel can be (and often is) booked a year before the departure date. It's not an absolute rule, and some agencies do offer “last-minute” trips, but there is some paperwork to be taken care of (insurance, medical check-up, vaccination) and most people seem to plan such trips well in advance so that some routes are already fully booked for 2015.

From South Asia, you have several obvious options to reach the US, often with a change of ship at some stage (you do need to plan for a week-long layover because the schedules are not guaranteed):

• Europe through the Suez canal (maybe from Jebel Ali in the UAE to Southampton) and then a transatlantic crossing.
• To China through the strait of Malacca (e.g. Colombo/Taipei/Shanghai) and then across the Pacific (e.g. Shanghai/Seattle). Or a direct crossing from Singapore to Houston over the Panama canal on a traditional cargo.
• Around the Cape of Good Hope and then across the South Atlantic to Brazil (e.g. Singapore/Durban/Santos) and then to the US (e.g. Rio de Janeiro/Houston or Santos/New York).

Those are just examples, there are many possibilities and most agencies should be able to piece together a route for you. You would be looking at something like a couple of months of travel in any case.

But the biggest problem is that I read in several places that India does not allow passengers to embark or disembark on cargo ships in the country. So, even though many ships do travel to India, you first need to go somewhere else like Sri Lanka or Singapore, quite far away from Mumbai, to catch a freighter.