Just like disk brakes, drum brakes also have a cylinder & piston assembly. Drum brake cylinders are smaller (less fluid used,) and double ended, whereas disk calipers have large cylinders, and only push in one direction.
It sounds like you've got a bad wheel cylinder on the passenger side. If it's leaking, and as oily as you say, you'll also need to put new shoes on, brake fluid contamination will ruin the lining.
Drums are rather complicated assemblies compared to discs. Half a dozen springs, linkages, cables, screw adjusters, etc. If you've never worked on them before, take both wheels off, and both drums, THEN ONLY WORK ON ONE SIDE. Use the other side as a model, remembering that parts are mirrored, i.e. don't think left & right, think leading & trailing components. You may need to pop the knockout plug at the bottom of the backing plate and collapse the adjuster if there's a ridge worn around the rim of the drum.
You definitely want to start hosing down the brake line connection into the back of the backing plate with rust penetrant a couple of times a day, for at least a couple of days before you start working on it, and use line wrenches, not open end wrenches to avoid ruining either the flare nut or twisting the line off. Personally, I'd think seriously about replacing both wheel cylinders, not just one. And I'd definitely replace both sets of shoes, and the remaining hardware is cheap enough, always replace it with a brake job.
