Why not simply staying with _PUTIMAGE? With the following syntax:
_PUTIMAGE (dx1, dy1), sourceHandle&, destHandle&, (sx1, sy1)-(sx2, sy2) 'portion of source to the top-left corner of the destination page
you may copy any cutout of the source image (sx1, sy1)-(sx2, sy2) to any desired startposition on the destination (dx1, dy1).
If you really wanna use _MEMGET/PUT, then you first need the _MEM handles of both images (your tilemap and your screen), then you have to use _MEMGET to get one pixel of the source and then _MEMPUT that pixel to your screen's _MEM handle, hence you have to use a nested loop for X/Y positions and run through all pixels of your desired tile. Also you have to correctly calculate the mem.OFFSET for each pixel, it's possible, but much more complicated than simply using _PUTIMAGE.