Istanbul Airport transfers to Istanbul hotels and from Hotels to istanbul Airport by reliable minibus and driver.
Istanbul Airport is 1 hour drive from the city. There are havas buses but they dont enter to old city Sultanahmet.
Sultanahmet is and old city from the byzantian time. There are lots of boutique hotels which will fit your budget. Also another point of the region is walking distance to all museum, mosques etc.
You need transfer from istanbul airport because the Istanbul airport is 1 hour drive from the city.
Catching a private transfer is easiest way to reach hotel in the city. ofcourse there are alternatives as catchig a taxi or catch airport bus till near your hotel region then change another taxi. that will be possible but you may loose lots of time till reaching your hotel