Is it possible to run a miner to confirm transactions for a target address? If I want to make a bitcoin transaction I need to wait 3-6 confirmations for my transaction to be accepted (by say an exchange etc)

Is it possible to run a miner so that it is listening out for transaction on my address, to have my transaction confirmed sooner?

Or at least have the first confirmation confirmed sooner?

(Plus would I be able to recoup some of the transaction fees?)
Yes, there are a few ways you can do this.

First, you could solo mine using GBT from Bitcoin-Qt, and prioritise your own transactions. But unless you find a block, this does no good.

Second, and perhaps easier, you could use a pool like Eligius which offers prioritisation as a perk to its miners.