If I start mining bitcoin today, can I mine the last bitcoin?
I don't know how old are you and I don't know how will the science advance,
but year 2149 is a bit far though, so the answer is no.
All in all, every ~4 years the block reward is reduced to half, so the miners receive less coins from the mined block. But you should keep in mind that they do receive all the transaction fees too.
and what will happen when the last bitcoin mined? What miners will do after mining all coins?
Normally the things should continue as they do now. Just the miners, while now they get X
BTC (from block reward) + Y
BTC (from tx fees), ... X will be 0, but Y may be bigger than now, at least as US dollars.
Maybe you want to read a little into
https://en.bitcoin.it/wiki/Controlled_supply