When I was younger and mowing my parent's lawn (with a fairly steep hill out back), I always just went across the hill, starting at the top and working my way back and forth and moving downward with each sweep across.
At the time I was using a gas mower that wasn't self-propelled, and I didn't find it much harder than doing the flat parts of the yard - you just need to be careful that the thing doesn't tip over toward the downhill side.
The only real downside is that when you hit the bottom, you then have to drag the thing back up the hill, but it's only once for the whole job.
The important mower characteristics are that it have a nice low center of gravity (keeps it from wanting to tip over too much), and that it have a working kill switch (modern mowers all have some kind of lever or bar that you have to hold in order to make them keep running).
Also, don't cut when things are muddy or wet - you don't want to risk slipping and ending up with a foot in the blade.
If you can't hold the mower upright as you walk across the hill, then my only suggestion is to rig some kind of winch at the top of the hill, and slowly drop the down the hill, then winch it back up for the next pass.
It (probably) won't suddenly stop and refuse to start again after 5 hours of use, but it's a good idea to follow the recommended cycle: the parts in new engines often haven't been milled perfectly smooth, so as your new engine turns those rough bits get knocked off or cut tracks for themselves, leaving little pieces of metal in the oil. Unless you get rid of them by changing the oil, they can cause excessive wear and tear on the engine.
Depending on your lawnmower, you'll either be able to drain oil out of the bottom, or tip the mower over and drain it out the top from the same hole you pour the fresh oil into (the worst part of this is lifting and tilting the mower over the catcher pan, but it's a lot easier if you have someone to help).
Best Answer
Sharpening the blade more frequently will make the task quicker and easier, since you won't have to grind as much to get a good edge.
If you do sharpen the blade using a power grinder heat from the process can weaken the blade, so you'll want to quench the blade to prevent overheating. dip the blade in water frequently during the sharpening process to cool it, and wipe it dry before you continue grinding.