3 elements for a small engine to run...fuel, compression & spark. Do you have test equipment to test for spark and compression? If not
Spark: You can remove the plug from the engine and hook it back up to the coil wire put the end of the plug near some grounded metal and pull the starter rope and see if you get a spark. If you see a spark chances are your okay in the electrical dept. If your not getting a spark it's not a fuel problem
Compression: While the plug is out put your thumb over the spark plug hole and pull the starter rope does it kick your thumb back 2 cycle engines need close to 90 pounds of compression to start & run 120 is more in the normal range. So this thumb thing is pretty mickey mouse but if you ge a real good push back chances are your okay there too.
Fuel: Is your plug getting wet when you try to start it? Remove the spark plug and pour a little gas into the spark plug hole? replace the plug will it fire then? If so you have a fuel delivery problem.
This unit is made by Poulan/weedeater, it's been my experience that there are a lot of problems with the gas lines rotting apart. Verify that the fuel take-up/filter which should be dangling inside of the gas tank is still attached to the fuel hose Is the primer actually priming? those hoses can rot out too. As far as removing the carb to clean it... there are probably some screws underneatth the air cleaner cover (by the way is the air cleaner clean?) If you remove those screws the carb will come loose you can then disassemble it...Chances are it's Walbro carb here is a site you can access... http://wem.walbro.com/walbro/family.asp
You can see exploded views of their carbs. One side of the carb is held on with 4 small screws on the corners, the other side, the throttle side;is held on with one big screw in the center, there is a diaphram under each cover. The fuel pump diaphram (the one under the throttle end) covers a couple of resivoirs where junk/sludge builds up and can prevent gas from getting through to the engine. If one of the hoses did rot out... debris from the rotted hose can make its way into the carb and blocks things up
But before I would suggest that you rip all this apart..be sure to check all of the simple things first.