Will be arriving into IST from Naples at 13:40 and will likely be at
hotel around 3-4pm. Departing next day to Singapore at 13:45. Well
aware that we don't have much time but will make the best out of the
situation
Hagia Sophia is open 9 a.m. - 7:30 p.m., you might be able to swing a visit on your arrival day if your flights and transport from airport are timely. The Blue Mosque closes around 6, so depending on prayer times you might be able to get there arrival day or first thing the next day. The cistern is open 9-7.
My last two cents, give yourself a generous amount of time to get back to airport and check in, the security there is very thorough including a security checkpoint right as you enter the airport (separate from security going to the gates see info here). Three hours ahead of an international flight is generally recommended (plus an hour to get to the airport, traffic is simply awful there) which doesn't provide you much activity time on your return flight day as many things aren't open until 9, except possibly Blue Mosque.
Good luck!