There are only two direct trains from Mumbai to Hospet (the railway station for Hampi). The Ajmer-Bangalore Garib Nawaz Express and the Jodhpur-Bangalore Express connects western India with Hampi. Both are weekly trains.
In other words you've two direct trains to travel from Mumbai side to Hampi. See the Garib Nawaz Express timings and days of operation.
Take the Garib Nawaz Express or Jodhpur-Bangalore Express (on Tuesday/Wednesday). Take one those return trains (Mondays/Saturday).
For both the journeys the trains are at odd hours.
Another strategy is to travel to Hubli by one of the many trains from Mumbai side. From Hubli, look a train or bus connection to Hospet/Hampi.
Another option is to get to Hubli ( UBL) from Mumbai. There are a few southbound trains from Mumbai that pass via Hubli. From Hubli to Hospet you can opt for a rail or bus connection. Hubli-Hospet is about 4 hours journey.
Another itinerary strategy is to go for a combination of flights and trains. Fly to Bangalore from Mumbai. Catch the Hampi Express (or any other trains) to Hospet. Complete the Hampi Tour. Take a direct return train from Hospet to Mumbai (or travel to Hubli and then to Mumbai).
Of course you can do this in reverse order too.
See also the Hampi Express ( Hubli - Hospet - Bangalore ) and Haripriya/ Rayalaseema Express (Vasco-Da-Gama - Hospet - Hyderabad/Tirupati)
There are quite a few bus services also from Mumbai to Hospet. You can book one of those sleeper buses, if train or flight is not an option for you.
VRL Travels, SRS Travels, Meenakshi Travels etc operate daily bus service (AC/Sleeper/Volvo) to Hospet from Mumbai.
The ticket price in bus hovers around Rs 1000 to Rs1500 per one way. This is somewhat comparable to the 3AC train ticket price.
(Please note that this inform could be changed with time and many to be up to date. You can easily check the correct details from on of the official websites of the Indian Railways at www.irctc.co.in or www.indianrail.gov.in )
Anegondi was the capital of the region, before it was moved to Hampi. In fact this was the core of a tiny kingdom that eventually expanded into the Vijayanagar Empire covering the whole of south India. Currently Anegondi is sleepy village with a principally farming community inhibiting it.