How can I remove this stuck nut on my riding mower?
See the holes through the pulley face?  Insert the heftiest screwdriver, steel rod, or what-have-you which fits and rest that against the biggest thing below it which does not turn.  Use that to prevent the pulley from turning.  Probably, you'll have to dedicate one hand to that while loosening.
Look carefully at the exposed thread.  The photo is not clear enough for me to be sure, but it looks like it might be reverse threaded.  So to loosen, apply force clockwise (as seen looking down the bolt).  Unless it turns out to be conventionally threaded, then counterclockwise loosens.
A ratchet and socket set like this will last decades.  3/8 inch drive is a good compromise between cost and durability/strength.  I still frequently use my Sears Craftsman hand tools accumulated in the 1970s and 1980s. The few failures I have had were cheerfully replaced (for free) years after I bought them.




For a nut so far down the shaft as that, a deep socket is called for.  Those can be purchased individually at larger stores.  Or sets of deep sockets at most stores, though that might be spending a lot of money for items you may never use.


If the nut was tightened well, the ratchet handle probably will not provide sufficient torque.  Simply extend its length by slipping a pipe over the handle, or attach Vice Grips, channel locks, etc. to increase lever arm&mdash;and therefore torque with the same pressure.

It could well be a two person job to initially loosen the nut&mdash;one to prevent the pulley from turning and one turning the socket handle with one hand and providing counter-torque on the socket with the other hand.  Counter-torque is often needed when the socket does not provide enough stability to keep the socket aligned with the nut.