Far east or west side Midtown from Times Sq you can find reasonable rates. Price will be around $30 per 24 hours.
They are all the same kind of lots, you drop the car off and they park it for you. If you tell them you park for 3 days straight then they will probably put it on a lift or somewhere far away. Tip the guy when dropping off the guy and tell him to take care of the car, works most of the time.
When picking up the car MAKE SURE TO CHECK FOR DAMAGE BEFORE LEAVING. Once you leave the lot you can't claim any damage.
If you want to park yourself I know of one in the area. It's on 42nd between 9th and 10th - http://goo.gl/maps/etmPw
But chances of being dinged there are much higher because people park their own cars.