There are many hotels and couple hostels in Almaty. The cheapest one I know:
I recommend you the 1st one - Hostel City as it is in the heart of Almaty. Easy walking to many sights. Just keep in mind Effeil Tower as orientation!
1)Hostel City a.k.a Almaty Central Hostel:
Address: Furmanova Street 189\2 apt.3, Furmanov street, (between Abay and Satpaev street, next to Eiffel Tower and bus stop. After busstop you will see an arc entrance to the backyard of the building. Go through that arc and turn left and you will see usual living building in the corner on your right. First door, apartment #3 if I remember. Opposite to that bulding - I think there is a police office ot whatever it is called.
Te.:+7 (727) 354-14-30
Skype: oligarxus http://hostelcity.kz/?q=content/nashi-kontakty
2) Kaz Hostel:
Address: 190 Shevchenko street, corner of Gagarin street
Address:145 Kurmangazi street, RC «Zhastar»
Cell: +7 -701-220-22-75, +7 -707-777-69-67