You definitely don't need a car in that area. I live two blocks away from the Drake and you can get anywhere in the area by public transportation, cab or walking (the weather is getting better daily).
Shuttle for $50 to the Drake? Take a cab - it will be quicker (no other stops) and actually cheaper (~$37 before tip).
The Westin on Michigan Avenue which SanDiego1K refers to is one block away from the Drake.
No real cheap parking options in the area, sorry to say.