Repair a lawnmower engine that was left exposed to the elements for a year?
Most common problems have nothing to do with your engine. Fuel, air, and fire are the key ingredients to your mower starting.


Fuel - if you left fuel in the tank for more than a couple of months without adding a fuel stabilizer, empty the tank and put new fuel in it.
Air - most mowers have an air filter. Look at it and clean it or replace it. It could also be the carburetor as that is where the fuel and air are mixed, but this would be the last thing I would look at.
Fire - spark plugs don’t last forever. Replace them every couple of years and make sure the electric wiring is making a good connection


Also, change your oil.

Mowers can be torn down for a thorough cleaning pretty quick (afternoon project). You’d need a socket set, flathead screwdriver, a can of brake cleaner, and nitrile gloves. Funnels and Mason jars are nice for this as well. The cost is really cheap: I lightweight tore down my mower last year and replaced the spark plug, primer, fuel line, fuel filter, and air filter for about $20.

Non-maintenance things that can also make for a hard/impossible start:


Bad primer bulb - normally will leak gas when pressed if bad
Pull cord and starter - never had this been a problem for me
Seized engine - if you can afford a new mower, you probably are better off