The Sokullu Pasa in Sultanahmet is a very decent hotel. I’ve used it 3 times. The first two times it was around $30 dollars for a room. The third time I showed up, it had been “Great Westernised” with much higher prices (and even nicer rooms) but they let us have a double with air-conditioning for $45 dollars.
It’s excellently located, no garage but the night staff can see the street and there’s a convenient lamppost to chain your bike to. It’s a beautiful old merchant’s house with an excellent courtyard for tranquil breakfasts.
Location: N41 00.233 E28 58.364
Kucuk Ayasfya Mah.
Sehit Mehmet Pasa Sk. 5/7
Try e-mailing them to ascertain a rate. firstname.lastname@example.org
Fax: +90 (212) 5181793
Maybe a little bit pricier than you’re looking for but it’s a nice treat while in Istanbul. There’s a much cheaper Sinbad Hostel nearby it but I don’t think they have a garage.