Renting a cheap car

I thought it would be fun to rent a car for a weekend or a week just to cruise around and get a feel for how Taipei is layed out. How much does this cost and what rental company would you suggest? I should use this credit card that comes with insurance. Is there anything like this Rent-A-Wreck we have in the USA that rents older cars? I should probably do this while I still have jet lag and can drive around at odd hours and not hit too much traffic.