While it may not be impossible, it would be pretty difficult to shoot a 1" group at 150 yds with 22LR. While the action of your rifle is OK (I used a 580 to build a 17HMR a few years ago, but it got a lot of working over), it is not as easy as taking a box-stock rifle and banging away. Several considerations-
* Rifle action- is the action capable of close tolerance work? If you can feel slop in the action, it is not. Many bolt action rimfire rifles may still shoot OK at 50 yds, but not so much at 100, and with a 30-year old 581 you are right out of there at 150 yds.
* Barrel- Factory barrels are lousy. Absolutely no redeeming qualities (as far as accuracy work goes). A good barrel could easily cost you $400-$500, and then there is still the work of installing it, chambering it, and tuning it.
* Triggers- Most factory triggers, like the barrels, suck. You need a trigger that consistently breaks clean. I have felt very few factory triggers on rifles costing under $1000 that feel good enough to take out for competition.
* Sights/optics- While a $68 Tasco may work OK at 50 yds, shooting a 1" group at 150 yds is going to take a bit more. Like several orders of magnitude better.
* Ammunition- and here's the kicker. Even if you have a match rifle that can shoot 1/2" groups at 100 yds, you may not be able to shoot 1" groups at 150 because of trans-sonic turbulence and the velocity of your ammo. Crossing the sound barrier will knock your bullets all over the place. Sure, you can start out with subsonic ammo, but you will have a harder time getting on target at 150 yds because of drop.
But how to do it? I can tell you it can be done. I built a 22LR rifle about 7 years ago that would shoot 10 rounds into 3/8" at 100 yds. It cost me over $750 in parts and took about 30 hours of skilled labor for the build. I am a gunsmith by trade, and my labor is free to me, and I like doing projects like that. I don't know how it would shoot at 150 yds because I never shot it at a target at 150 yds, but I shot at 100 yds quite a bit for about a year. I can tell you that it was a one-holer, and I can also tell you that I will do it again soon. Yes, it is fun.
Good luck with your project!