Can the back axle be replaced without removing the gear cassette? I'm about to replace a back axle for the first time, and I'm confused about whether I need to remove the cassette to do this properly. This is relevant because I don't have the tools to remove the cassette, so I'm wondering should I hold off on the job until I have those tools.

Here's the issue. My back wheel uses a cup and cone bearing system. I understand how tightening the cone works and I have all the tools, but the problem is that when the axle is inserted the cassette side cone is concealed by the cassette, and so it can't be tightened when the axle is in.

My idea for getting around this is to fix the cassette side cone on the axle using the associated nut while the axle is out, then put the axle in the hub, and then tighten the other cone and nut to finish the install. Is this reasonable?

(Sorry if this is a stupid newbie question, but neither my Zinn book nor the online articles and videos discuss the relationship between installing the axle and the cassette.)
It is possible to do this, at least with some cassettes/freewheels, either in an emergency where you don't have the right tools with you (but see here for a small cassette tool), or where you can't easily find the right freewheel remover for an old bike.

If you are at home, it isn't urgent, and you have a modern freehub, don't bother with this, just buy the proper tools first.

Use a cone spanner and normal spanner on the non-drive side to loosen that locknut, then remove the locknut and nut. If necessary (because the drive side locknut undoes and you can't reach the cone), hold the axle with a vice, or vice-grip wrench - you will chew up the thread, but you were going to replace it anyway. (If you have to reuse the cone and locknuts, undoing them over a chewed up thread may damage their threads, so try to avoid that.)

(If the axle is broken, you don't have to undo anything, just pull half out each side.)

Withdraw the axle and drive side cone from the cassette (this is where @Daniel's "in some cases the cone may be larger than the hole in the cassette" may make it impossible).

Clean the cups - it is a bit awkward doing one inside the cassette, but not impossible.

Tighten the drive-side locknut and cone against each other, hard. Insert the axle. Do up the non-drive cone slightly tighter than you want it to end up. Do up the locknut, then tighten the cone and locknut against each other. If it ends up wrong, undo, adjust, and repeat. It is tedious to do this without being able to use a cone spanner on the drive side, but it is possible. (You can put a spanner on the drive locknut, but if it becomes unlocked, you'll have to take the axle out and start again.)