Good advice above. Check the easy things frist-filters, plug, carb kit and so on. As far as rebuilding it, the 044 is a saw very much worthy of a rebuild. If you've read any of the homeowner vs. professional model threads, you'll consistently see the rebuildability of the saw mentioned. If you're going for a basic rebuild, a new jugs and pistons are available from a dealer, Bailey's (but they aren't factory Stihl if that matters to you) and klickitatsacket here on this site. But then again, it may just need new rings, and that would cheaper and easier. Now, there are some small differences between the older 44's and the new ones. One is the wrist pin. The newer ones are lager, and I can't remember the size, but it's something like 13mm to 15 mm or something like that. That would be a consideration if buying a new piston-you'd need to get the right one. I've got the P/N's around here somewhere. I'm moving right now, so it's packed away, but if I can find it by the end of the weekend, and they haven't been posted yet, I'll get them on here for ya.
If you're just a weekend guy, I'd say it'd be hard to justify having the saw built for anything beyond stock use. It's plenty for you. You can get a factory dual port muffler cover for it during the rebuild/overhaul. I think the P/N is 1128 140 0801. Again, my pubs are packed away right now, so I'll check that for you and get back if that isn't it. That muffler cover is a factory, bolt on part that'll add a noticable increase in power for you. They should be about $30 from your dealer.
But yes, I'd rebuild the saw if it's in any kind of good shape. It's a good saw. The 361 is a fine saw as well and will run you about $570 or so. A rebuild for your 044 will be substantially less, especially if you do it yourself. Do it yourself, and you can rebuild the whole thing with factory parts for less than $275-$300 easily. Even less if you go with the jug kits from Bailey's, as they're only $99. If you have a shop do it, it'll be that plus labor.
Welcome to AS.
Jeff