Its very hard to recommend someone,some place at my age as long as all is not under your control...I feel as if it is a responsibility when something goes wrong..So some tips first then links recommended by bikers I ve met here...
Most bikers stay at cheap hostels while they have to be close to consulates for visa purposes.(except,tajikistan,turkmenistan and kazakistan--these are around Yeşilköy Atatürk Airport)
Sultanahmet area is touristic İstanbul and very expensive.. Sirkeci,kumkapı,cankurtaran are the parts of real istanbul close to Sultanahmet.
(about 500-700 mt.uphill walking distance)
Traffic always problem inside the city so best way to get there is to take the seaside road (Kennedy Rd)via connections from E-5 before or after the airport and follow aksaray,eminönü road signs and lines and take the sultanahmet turn at Cankurtaran..
Best places for meals (price and quality)are local doner buffets around (not inside) spice bazaar.Ones next to Big Postoffice(Büyük Postane) on the way to spice bazar recommended..
Opposite of St.Sophia, there is a place with metal bars under a tree where you can park and lock your bike..(sure without panniers and valuables on even is most crowded place 24/7)
Next to the hamam on the left corner of the park while walking from St.Sophıa to Blue Mosque,there is a private parking area 24/7 with guards.
In front of Blue mosque, there are many wooden sofas to sit and watch the laser and sound show every night and many tourists spend the night there free..
Ramadan here and some government organisations available at Sultanahmet and Beyazıt squares for dining and amusement..Meals given at 8,45 pm. everyday under huge mayor tents or tabled areas are free.Sure very hard to find a seat in crowded places.
Here are links to 2 of the hostels which are only for foreigners and have also dormitories....They generally find local solutions to parking problems..This week, will be at the city center and will search for the ones with parking availabilities..
Rooms - istanbul Harmony Hostel
istanbul piya hostel, sultanahmet, istanbul cheap and budget hostels, istanbul rental apart rooms
All the best...