I would cut open the drywall as mentioned in a previous post.
That way you can see exactly what you're dealing with (iron or copper piping) first before you shut off the water and decide on your gameplan.
Also, if you don't know what pipe in your basement or crawlspace supplies water to this hydrant, it might be a good idea to install an access door on the wall and have a shut off valve upstream of the hydrant which you can use the shut off the water supply to the hydrant for repairs.
Mifab makes a line of access doors and they can be ordered from any plumbing wholesaler for anywhere from $20 to $200, depending on what you need. They make access doors to fit stud spacing of both 16 and 24 inches, both with and without locks. A simple access panel to cover the hole shouldn't cost any more than $40.
Roof, Floor, Area, Trench, Drains Carriers Lav, Toilet, Interceptors Hydrants TSP WHA - MIFAB - Drainage and Plumbing