If there is a fair amount of soft black carbon in the combustion chamber and exhaust port, the previous tenant may have been using too much oil in the fuel, and/or letting the engine idle a lot. However before letting it go at that, I suggest that you check the compression with and without a small (tea)spoonful of oil put into the combustion chamber through the spark plug hole. If the compression increases substantially with the oil in place, chances are the piston rings are worn out or damaged. Also, take a look at the crankshaft seal to see if it is leaking (The 2816 appears to be a cantilevered crankshaft design, with only one seal).
If one or both of the diaphragms are stiff and/or have wear marks, the carburetor needs an overhaul kit. So far I see no reason to suspect a problem with the mixture needles, which really only happens if the previous tenant has played with them.
Essentially, most whipper snippers are disposed of because they have become difficult to start. Sometimes the reason is that the spark plug has deteriorated or dirty fuel has blocked a fuel filter (in tank or carburetor) or less often, a jet or passage. In the remaining cases, either an incompetent tenant has messed them up (and there are usually visible signs of this) or a possibly competent tenant knows very well that they are worn out. You go through a process to find out which of these three failure modes has occurred.
