There are no downtown hostels. A number of people in the city are talking about attracting one in the near future, and it would be great. There is also an exciting innercity rail plan being worked on that would make Cleveland a major hub. I hope these plans will come through. I love backpacking.
The best budget hotel in the city of terms in price with an EXCELLENT location is the Holdiay Inn Express downtown. This hotel also has a bit of beautiful historic character. It was once an old bank. The facade and lobby look really cool. It is priced around $70 though. The Comfort Inn downtown is also similarly priced. There isn't much I can find on the web downtown for under $70.
If you are backpacking via Amtrak, Airport, or Greyhound you should know that Cleveland does have a mass transit system comprised of light rail and bus that would allow you to get around the city without a car. The city's major transit hub is in Terminal Tower/ Tower City. The Amtrak and Greyhound stations are located downtown and within fairly reasonable walkind distance to Tower City, and the airport is linked via rapid red line. There is a costumer service desk there and brochurs on all the routes. All the light rail stops in the station here, and outside of Terminal Tower on Public Square over 80% of the county-wide busses go to Public Square. It only costs $3 for an all day pass on all Cleveland public transit. So this is certainly an option for finding a cheap hotel, but connected to the city center.
I am from the west side, but checked out a map and the Shaker House Inn (South east of downtown) is at the terminus of the Vank Aken line, which is the line that runs to Shaker Square, downtown (Tower City), and the Rock Hall. This is actually a pretty nice area to stay in (beautiful historic and famous "garden city" suburb built in the 1920's, and you have to stop at Shaker Square for a visit).
The West Side alternative to Shaker House Inn would be the Days Inn or Travelodge on the border of Cleveland in Lakewood (Cleveland's Gold Coast and the highest density city between New York and Chicago). It is in a pretty nice neighborhood with a lot of character, and around the corner there is a grocery store and some neat neighborhood restaurants, bars, and 24hr diners on Clifton Boulevard. Rates are around $50. Just north of these hotels on Detroit Avenue and W.117th there is a nice little agglmoration of bars/restaurants/diners. There are also a number of bars/restaurants located on Detroit and Madison Avenues in Lakewood. Lakewood is served very well by mass transit, and Clifton boulevard bus routes in particular will take you downtown in a matter of minutes.
Check out the "airport" hotels too. A lot of them listed have 24hr shuttle service to the airport which is another light rail transit stop in Clevleand that runs directly downtown. A lot of these hotels are within walking distance to the airport terminal or the Brookpark Transit station.
If you like backpacking, which I do, you have to stop at the West Side Market (Ohio City stop), Shaker Square, Tower City, E.9th Pier, and University Circle. From the University Circle and Shaker Square stations you can hop on the 821 bus circulator that connects a lot of cool neighborhoods in Cleveland like Little Italy, Coventry, Cedar-Fairmont.
There is one hostel by the way in the Cuyahoga Valley National Park. You could go via bicycle on the Towpath Trail, but it would take a while and you would have to find a bicycle.
Feel free to add comments with additional questions or clarification.