Go to gunbroker.com and check around for it. Try to match it up with models in the same year and condition. That website is usually a little on the high side of price for used guns but it will give you a general idea.
Not a common caliber anymore.
22 Remington Special is almost identical to 22 WRF. Neither of those are the same as 22 WMR, which is normally referred to as "22 Mag".
So, your rifle won't accept 22 Mag. It's chambered for the shorter 22 WRF (22 REM Special).
So, for value, it's probably more valuable to collectors than shooters. As for selling it in the next 36-hours, that time-limit probably won't help you get the best price for that rifle, in that caliber, in that condition.