Transportation

It is not advisable in some country to take an unofficial Taxi, but in KZ it is ok (generally). Even at my local Hypermarket they sell the Taxi sign.  Just use your common sense, don't pay upfront and keep your wallet out of sight and split your money on you.
You can negotiate on the fare it all depends on how much you are willing to pay i.e. 1,000 KZT -2,000 KZT from airport to  city centre. Further the sky is the limit.
There may be long distance coaches/buses/train that will take you to your destination, but as I don't live in Astana, I don't know which one or any. You could search and check online.
If you are claim expenses make sure you have a receipt/check (stamped and signed for) as Kazakhs love their paperwork and official stamp here. Without them they may not pay you back your outlay.