NO!
Wood easily splits or shatters, and then the broken ends act against each other as wedges. So, besides the stuck bullet and/or case, now you have a dowel stuck in the bore.
To make matters worse, this is .22 caliber. That means the most commonly available wooden dowel size that will fit is 3/16" (0.187").
A 3/16" dowel will crap all over your world, if you try to force an obstruction or a stuck case out.
The larger the dowel size that you can use, the lower the odds of breakage. However, .22 caliber is pretty much all the way at the other end of the spectrum. It's about as small as you can go, before the "dowel" is a toothpick. DO NOT use a wooden dowel to push an obstruction or stuck case out of a .22 caliber barrel.
Most people have steel cleaning rods. They work, but are not recommended.
Brass rod (360 or harder) works well, but most people don't have them just 'lying around'.
Nylon may work, as long as it is
clean* and you're using the rod's weight more than forcing it. Most people don't have nylon rods handy.
I, personally, use O-1 steel drill rod. Sized properly, or purchased undersized and given a wrap of electrical tape to prevent contact with the barrel, it won't damage the crown or lands.
Depending upon where you buy it, lengths are available from 3" to 96", in number, letter, metric, and decimal (non-standard) sizes. You can get pretty much any diameter, within +/-0.003", in 3-foot and/or 4-foot lengths for less than a cheap, ill-fitting, steel cleaning rod.
If you do buy a piece of drill rod, file any burrs off of the ends before use. And if you EVER tap one end with a hammer, mark it so that that end never goes into a barrel again. (I put a big wrap of electrical tape around it, and mark that end with red paint.)
I do a lot of "experimenting" with hand loads, and non-standard loads, so I keep at least one piece of drill rod (at least 30" long) for every bore size that I own, except .475 caliber. To me, they're worth their weight in gold. Often, the weight of the rod is enough to break a bullet free and push it out of the bore - especially if I oil the barrel first.
(.22 cal [3 sizes], .24 cal [2 sizes], .27 cal, .30 cal [2 sizes], .32 cal [2 sizes, plus the .30 cal], .35 cal, .44 cal [2 sizes], and .45 cal -- I just haven't gotten around to ordering the rod for the .475.)
If you don't want to buy a piece of quality rod (brass rod, drill rod, whatever) for this one fix, then just let a gunsmith take care of it.
DO NOT try to push barrel obstructions out with wooden dowels.
Many people do it, and many people recommend it. But, eventually, you WILL have a dowel shatter and get stuck in the bore.
Something to think about: When was the last time that you had a gunsmith recommend using a wooden dowel?
I used to use dowels. ...There's a very good reason that I went to O-1 drill rod. It involved a stuck lead slug, a dowel (cut into 6" sections), and the
month that it took me to pick the wood splinters out. ...And then I still had a stuck lead slug to get out.