Hey Nick, how ya doing?
The sandpaper would work, but it's not a choice I would make. You say you have a small stone for the axe, what kind of stone is it?
I would suggest just practicing the old free hand sharpening with that stone. It's easier than you think. Main thing is not to think about it too much, because if you over think it, it becomes over complicated. Don't worry about angle and micro bevels and all that stuff. Get yourself a dry erase felt tip marker to color the blade edge with and go to work.
Take the knife, and with the stone laying on a table, make like you want to cut the stone in half, with the blade contacting the stone surface at 90 degrees. Holding it there on the stone in the middle, lay the blade over half of that. This puts you at about 45 degrees. Now cut that distance in half again, and you are in the neighborhood. Start honing the blade edge in a small circle movement, keeping in contact with the stone while honing. Do this starting back at the kick, (base of the blade near the handle,) and slowly take a full minute to work your waay to the tip of the blade, keeping in contact the whole time. When you get to the tip, turn the blade over and do the same thing on the other side. Repeat as needed. Use the felt tip marker to keep track of what your doing. After a while, your kife will have a pretty good edge, and you will develop a feel for it using less pressure as the knife gets sharper. A stropping on leather after this will give you a shaving edge on your knife. This is the old way the boy scout manual had to sharpen back in the 1950's. It works fast, anyone can do it, and it's effective on a wide variety of mediums, like the bottom of a coffee mug, a piece of brick or smooth stone, or pocket size hones.
I've tried all the wauys to sharpen a knife, and I even went the diamond route. I eventually went back to what worked grat when I was a kid. Sometimes the simple old ways are still the best. It will let you resharpen literally anywhere, at any time. It ain't rocket science, and don't waste your money on fancy gizmo's. A small pocket stone and practice is all you need.
Carl.