you can get in one day if you go in person in London,must be there by 9.30am latest as they only accept so many people, big queues, little space and too few seats, take a book or newspaper.Download forms from internet, dont forget your photos and you need cash to pay.this can take several hours and they ask you to collect after 4pm also chaotic.
Alternative is to get an agency to do it for you, worth the money no hassle.
Word of warning, last year I was in the queue with a very elderly gent and they had lost his passport and he had sent by reg post amonth before, they were very rude and very unhelpful and disrespecftful to such an elderly person, he got nowhere and was due to travel next day.
Friends in 2005 sent their passorts [ a married couple] by reg post when returned there was a third passport with theirs, they got in touch with the next of kin at the back of this passport and a guy drove down from the midlands to Suffolk to collect from them, imagine if they had sent it back
to the Embassy they would have just kept it and not had an address for it[ I am sure they would not have bothered to look at the back].
No you cannot purchase on arrival they will send you back next flight, but I doubt they would let you board your flight anyway, you would be turned away at check in.
Its getting very hot in Goa now and a lot of the tourist things are closing as it is end of the season. I think the last charter will leave soon, make sure you can get back otherwise you will have to get to Mumbai to get home.