The main things that you are going to want to be easy are things like changing the oil, greasing and changing out belts. If you are tinkering with your machine every day to make it run without problems, there is something wrong. You should not have to be doing repairs on it regularly.
At the beginning of my day when I'm mowing (before I even start the engine) I'll check the oil level and make sure that the air filter is still okay, I'll glance at the tires to ensure no flats and check the hydro level (which for me is just a glance at the tank since mine is transparent).
As far as the maintenance I do, I change the engine oil every 50 hours. I change out the air filters as needed. I grease components every 50 hours (per my owner's manual). I change the blades on the mower every 10-15 hours, at that same time I clean under the mowing deck and just do a visual inspection of the deck belt. I also use my leaf blower and blow off the mower at the end of every day so it gets put away clean. There are a few little things like changing the spark plugs and fuel filter, which I do annually.
I do bring in my mower to the dealer for hydraulic service (which hasn't been done yet as my mower is too new) and engine repair (again my mower hasn't needed any repair).
You don't have to be a mechanic to own one of these mowers. I really am not that mechanically inclined, I can do the general maintenance stuff but thats about it. But my machine stays in great shape both cosmetically and mechanically.
As far as buying a mower, the 300-500 hour range you mentioned is not a bad range. These mowers will last 2000+ hours with proper maintenance. You can buy new, which makes it nice since you'll know the whole history of the mower but buying a mower with 300-500 hours on it will save you thousands. I would shy away from mowers with over 1500 hours if you aren't mechanically inclined. You best bet, is to stay with mowers with under 500 hours.
'07 Chevy 2500HD
2010 John Deere Z925A 54" MOD
2011 John Deere Quik Trak 647A
John Deere walkbehind