Sirkeci Station is a five-minute walk from the hotel and worth the detour even if you are not taking a train. Opened in 1890 as the terminus of the Orient Express, it still has its original hall, stained glass and platform clock.
Inside there is a free railway museum, open most days, and a restaurant on the platform that dates from the station’s first years. The Marmaray line runs from the lower level, which is also the quickest way to cross to the Asian side — useful if you are heading to Kadikoy.
Twenty minutes is enough for a look. The station is on the natural route between the hotel and Eminonu, so it fits into a walk you are probably making anyway.