Wow, that's a tall order, folks!
Oleg, it really depends on the bore dimensions, the action length, the action STRENGTH, how much meat is in the barrel at the chamber end, and the bolt face.
For example, take Tamara's 7.7 Arisaka. There's a basic Mauser-type action, with that goofy thumb safety, firing a rimless round with a .311-.313" bullet. The action is plenty strong, as evidenced by old P.O. Ackley's famous experiments in which he purposely tried to blow up an Arisaka using the wrong caliber ammunition.
Tamara stated she wanted to know what her 7.7 Arisaka, with sucky headspace, would "clean up" to. I have to assume she wants to use the original barrel with it's .311-.313" bore. The first thing one could do is remove the barrel, take it back a thread or two, and run a 7.7 Arisaka chambering reamer back into it, thereby creating a new chamber for the original cartridge. When reassembled onto the receiver, the biggest difference is that the barrel will be a fraction of an inch shorter than original.
Since it's also a .311-.313" bore, she could have the original gun barrel set back and rechambered, to the Soviet 7.62x39 M-43 round. The bullet diameter is proper for the Arisaka's bore, but the Arisaka's claw extractor would need to be bent inwards just a bit to get a better bite on the smaller diameter 7.62x39's rim. Obviously, more than a few threads' worth of barrel set-back needs to be done to accomodate the shorter round. The benefit here is the cheap surplus ammo, and it would be a real pussycat to shoot the full-size Arisaka with the 123gr 7.62x39 ammo.
Sticking with that .311-.313" bore factor, the Arisaka could also be rechambered to the 7.62x45 Czech M-52 round. More oomph, but the problem here is availability of good non-corrosive ammo. I handload this round, but only because I have this weird affliction of collecting Czech VZ-52 SHE rifles. Not a real good idea for converting the Arisaka, but it can be done...
Same goes for the 7.65 Mauser, first used in the Belgian Model 1889 rifle. Again, while bullet diameter is proper for the 7.7mm bore Arisaka, and the 54mm-long 7.65 Mauser offers more power, it's a question of where are you gonna get good ammo for it?
Many GI bring-back Arisakas were gunsmithed to have their chambers reamed using a .30-06 Springfield chamber reamer, leaving the bore dimensions as is, to create a .31-06. In other words, .30-06 brass was handloaded using the .310-.313" diameter bullets intended for the .303 British and other .31 caliber rounds. This combination has been around for quite a while, giving GI's the means to shoot their wartime acquisitions, and there was plentiful .30-06 brass to be had, as opposed to the 7.7 stuff, now made by Norma. The problem with this approach was that the original 7.7 Arisaka's chamber was a bit larger in diameter than the .30-06, so if the original chamber was cleaned up to give a .31-06 without setting back the barrel several threads, the fired brass from this hybrid would swell at the cartridge base. Depending on how loose the 7.7's original chamber was, this could be icky at worst, or no factor at best.
What about that .303 British? Well, the bore dimension is ok, but the British round is rimmed, to the tune of about .540". The bolt face would definitely have to be modified to accept this big rim. Same for the extractor, in all probability.
The best bet for the Arisaka, if you're not gonna rebarrel it to a different caliber, is to leave it as a 7.7 Arisaka chambering, and either make handloads using either cast bullets or the .31 caliber jacketed offerings. If the chamber is too far gone to offer any accuracy, then I'd suggest either setting the barrel back and rechambering to 7.7 again, if not 7.62x39. Or, since the barrel's off the action, maybe a .308 Winchester or .30-06 Springfield barrel could be fitted, giving the owner a gun they can shoot cheaply. Mounting the original front and rear sights would make the rebarreling a lot of fun, though.
These "fixes" are seldom cost-effective. While I've seen many beautiful custom guns built on the Arisaka action, it's days as a donor action for fine sporting firearms has waned. Last time I talked to Dayton-Traister and Timney, they no longer offer an aftermarket adjustable trigger for the Arisaka. My mom has a custom Arisaka, the earlier 6.5mm variant, with a Bishop maple Mannlicher stock, 22" bobbed barrel, deep nitre bluing, aftermarket trigger, low-scope bolt and safety, and a nice Redfield variable. It's her deer rifle, it's graceful and fun to carry and shoot in the original 6.5x50 Arisaka chambering. I doubt you'd find many folks willing to do the same kind of work on an Arisaka in this day and age.
I have a 6.5-06 long-range Interdiction rifle, built using a Krieger barrel on a 1916 Haenel Mauser 98 action. It shoots exceptionally well, I have over $1,100 sunk into it, and it's not worth the sum of it's parts to the person who is not aware of the gun's history. But it was worth every penny to me when it won several matches, to include centerpunching a golf ball at 500 meters.
Some guns do much better when rechambered to a commercially available caliber. The 7.5 French M-49 semiauto rifle was never an easy gun to keep fed. Since then, there's been a bang-up business rechambering the guns to .308 Winchester. So much for collectability, but at least the owner can shoot them on a regular basis now.
So, how confused have I made you with all this, Oleg and Tamara?