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I would strongly suggest getting a place close to your office, because public transport is a little disorganised here. Maybe stay in a PG for a month or so while you hunt for flats, don't sign a lease without looking at the place because the construction quality in Town can be extremely unpredictable.

The Local serves the same purpose as the metro in Delhi, yes, but you also have to keep in mind that the network and the trains and the stations are decades older. So temper your expectations accordingly.

Anyway, if you're living in Town (South Bombay), it's unlikely you'll need to take the local to travel to office every day. It's mostly kaali-peeli cabs, which serve the purpose of autos in Town, since entry of autos is prohibited (I take it to the office whenever I'm running late, they run on metres and the fare is 40-50 rupees for 2.2 kms) or buses. I personally really like the buses, the network is very widespread and it is very easy to find a station near wherever you are. The only issue is, the traffic makes them a little unpredictable. So I usually take them only when I have a margin of 15-30 minutes. Or I just walk. 2kms isn't that much.

You will find several PGs around Fort area and Marine lines, a lot of colleges and offices are here, so the immigration rate is high. Since you are coming for a longer term, you can try looking for a flat as well. But the absolute lowest you can expect for a 1BHK is 30K and the lowest for a single occupancy room in a PG is 20K. Food extra. Even with flatmate(s) with everyone getting individual rooms, don't expect anything below 20K.
Mumbai is a very broker-heavy city, so your best bet is to look online or get in touch with some friends who know brokers. Maybe ask you office colleagues for some leads.

Helpful apps:
mIndicator (a guide to all modes of public transport in the city, vv helpful)
UTS (for booking Local tickets online. The ticketing queues are miles long)
Chalo (UTS for buses. The queues aren't the issue here, but it comes in handy if you don't have change)