Hey welcome to the wonderful world of taking your life into your own hands and giving the MAN the bird !
I have a shop manual for this saw...but it is mine...and I don't think it would help you anyway.
However I may be of some assistance to you if you have specific questions. I have rebuilt my 290. Ask me anything and I can try and help.
I would make some recommendations:
One, upgrade to the 310 cylinder and piston...or the 390 [for which you will need to deal with a decompression button issue...but it is possible]
More power is usually better...[opinion]
Next realize that this undertaking is not rocket science and that the MAN will tell you otherwise...that is what keeps him in business.
Make a concerted effort to find the ROOT of the problem...it makes absolutely no sense rebuilding an engine that will just experience the same catastrophe.
If you can not figure it out...AT THE VERY LEAST replace the impulse line, the crank shaft seals, and the rubber gasket between the carborator and the engine....these are your most likely culprits of and air leak that contributed to the ultimate failure. [assuming you did not make a stupid mistake like running gasoline without oil mixture, or dropped something in the engine while cleaning, or running full throttle all day long, etc...]
Please let me know why your engine failed...as I am very interested in any chainsaw antics.
It is going to be difficult to get the rings on the pistons...and make sure they line up with the pistons' "clit". If they do not when you finally insert...oh ####!
BUT this is pretty well done with your god given fingers. I think Bailey's sells a ring clamp to help..but I bet it sucks for $5...who knows may be you should give it a try and let me know.
Use the proper sealant when rebuilding the cylinder...not too much...but enough to JUST over-fill the indented bead. The good stuff is BLACK in color, the same used for any cylander...you can get it at the auto parts store.
Finally,
Break it in slow and easy. Let it blubber in 40:1 [nice and oily], not 50:1...if you really care for this saw and all your hard earned work. Then cut some wood, keeping it under load at HALF throttle....do this for 3-4 tanks of gas. Then go ape ####.
And always remember that you have done what the MAN despises...and pass on your knowledge to the next...and so on.