What's the easiest way to polish a barrel?

I would just use emery cloth. I think start 4000 and work you way to 8000 grit.
depending on how comfortableyou feel doin it, you find a plastic or wooden rod that long enough for your barrel, cut a small slit at the bottom of the rod so the the edge of the paper can slid in. Then just wrap the emery cloth around until it the right diameter for you barrel.
Now you can do this by hand or if your careful you can an electric drill or something. But you just run the rod back and forth in your barrel and turn it at the same time.
Also on yhetherod you should put some sort of stop so you can go to far and damage o rings.
Anyhow that how I did a little polishing on mine. Except I used the lathe to spin the barrel much easier that way.
 
If you use sandpaper and spin it all you are going to do is hit the tops of the lands and do nothing to the groves... unless you are making a smooth bore, haha.
You need to look up 'fire lapping' and do that with abrasive grit.

For little polish which is not as aggressive as fire lapping, I do use JB Non-embedding Bore cleaner some times, put some on the pellets nose and some on the skirt but none in the base then shoot about 20 to 25 like that and it will do a little polishing. I have not found the cleaning pellets of much use and consider them a gimmick because there is not enough side pressure to scrub the side against the barrel all the way down. AND I have seen baffles damaged by the cleaning pellets as they sometimes turn sideways leaving the muzzle and gwt stuck, then the pellet comes along and hits this obstruction and... guess what you are fixing/replacing?
Heads up - if you shoot the JB through your baffles you will need to clean them, yes? {and it gets into shrouds with air strippers too}

NOTE: It only works on a cleaned barrel because if you have lead streaks already there all that will happen is the lead is not removed and in fact gets worse buildup. Lead wipe-away cloth patches are great for deep cleaning a crudded up airgun barrel, run the patch on a tight jag back and forth 5 or 6 times in the barrel, flip sides and do again, change the patch and repeat. (a single pack of the cloth last dozens of guns a least a year or one gun about ten years)

After the deep clean I use Ballistol (mixed 1:4 Ballistol:water) to clean out that residue and lube the barrel. Then I use the JB to do the polish and clean that with the Ballistol again. Fire about 50 conditioning shots and you are good to go.

And remember: You can't make a silk purse out of a sow's ear... if the barrel is junk to begin with nothin anyone does is likely to fix it. It has to be halfway decent to polish and get any significant improvement.
 
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 I did it thank you everyone for the information.. here my results the ones out of the group was because of the wind, the targets was at 55 yards 
 
Please pardon me but suggesting someone use sand paper on a dowel with an electric drill is , mean? More likely just a bad idea offered.
Personally I would pull the barrel and use a name brand polish with a name brand polishing kit. Just one brand:
https://shop.brownells.com/gun-cleaning/solvents-degreasers/bore-cleaning-paste/index.htm?cm_mmc=PPC-_-Itwine-_-Google-_-083-065-100&gclid=EAIaIQobChMIv8Xs9s-S2gIVUoZ-Ch3ByQvkEAQYAyABEgLJUfD_BwE

And if really not wanting to pull the barrel, "Patchworm" works well. Fire lapping can work well, it can also take an astonishing amount off in rare cases, if fir lapping do make sure to thoroughly clean ALL grit out of the barrel. 
Fire lapped a Carrer 707II once then an honest hours cleaning it after that and in that case it really worked wonders. I used the heavy korean ammo ( holds lot's of goop) and fired three of those, cleaned, happy.



John
 
I used this 
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spysirPlease pardon me but suggesting someone use sand paper on a dowel with an electric drill is , mean? More likely just a bad idea offered.
Personally I would pull the barrel and use a name brand polish with a name brand polishing kit. Just one brand:
https://shop.brownells.com/gun-cleaning/solvents-degreasers/bore-cleaning-paste/index.htm?cm_mmc=PPC-_-Itwine-_-Google-_-083-065-100&gclid=EAIaIQobChMIv8Xs9s-S2gIVUoZ-Ch3ByQvkEAQYAyABEgLJUfD_BwE
And if really not wanting to pull the barrel, "Patchworm" works well. Fire lapping can work well, it can also take an astonishing amount off in rare cases, if fir lapping do make sure to thoroughly clean ALL grit out of the barrel. 
Fire lapped a Carrer 707II once then an honest hours cleaning it after that and in that case it really worked wonders. I used the heavy korean ammo ( holds lot's of goop) and fired three of those, cleaned, happy.

John