I shot Bullseye for a number of years and got fairly decent at it, but I was shooting 2000 rounds of rimfire Bullseye and 1000 rounds defensive shooting a month. The Ruger you have is an OK shooter but you'll find that the air gap on the front sight is excessive for good Bullseye shooting and the trigger is rough. A drop in Volquartsen trigger will be a lot better than the stock Ruger trigger (I have found Ruger triggers to be gritty and inconsistent) but even the Volquartsen trigger is not even close to a broke in Mod 41 trigger. I would also look for different width rear sight blades if you stay with the Ruger for a while. If you are serious about being competitive with Bulleye shooting, drop the bucks on an S&W 41, Volquartsen, Hammerlli, Walther or one of the other top end Bullseye guns on the market. A good bullseye gun is an investment, but assuming you have the the shooting ability, they will raise your scores significantly. A good used 41 will run between $900.00 and $1200.00. If you are just going to have fun with bulleye the Ruger will be fine.
One of the biggest influences on rimfire accuracy is ammo (once the trigger, sights, form and timing is worked out). My 41's will shoot most ammos fairly well, but really come into their own with particular brands of ammo. I advise getting as many brands of ammo as possible and find what ammo your gun likes. I used a C-More dot sight and sand bag rests testing ammo, shooting 5 groups of 5 rounds for each brand, after I found what the gun seemed to like, I shot them more. Try some of the target specific ammo and compare group accuracy to the standard brands as I found some of the target ammo did not shoot any better than standard ammo in my guns. Each of my 41’s both shot Green Tag and Federal target ammo very well (ragged 1 hole groups over bags at 50’ if I was on my game). If you get really serious about it you’ll want to purchase ammo by lot numbers as there can be significant differences between lot numbers in how the same ammo shoots. I used a rim gauge dial and sorted cheaper ammo by the read out and could get Winchester Wildcats to shoot fairly well for practice ammo, abet there was more flyers.
Good luck and enjoy, Bullseye is great fun…and Bullseye shooters have almost as many excuses as Trap Shooters…. in the end it always the damn guns fault.