IMHO 7 hours is not really enough to get to the Old Town. Traffic is bad (if you take taxi), metro takes up to 1 hour and to get to Hagia Sophia/Blue Mosque you need to change for tramway.
Even worse, the seven hours would shrink to 5 after the arrival faff and giving up tine to reenter the system and get to the gate. But if that allowed me only a couple of hours to waltz around Sultanahmet I'd be happy to get out of the airport and into the fresh air.
Now serious business: we went from no Istanbulkart machine at the airport to (I think) one, and occasional long queues with tourists on steep learning curves: are there any more machines, or are any of the retail outlets selling the card?