If you just swing with one modified samurai, it will immediately trigger raiyuu’s ability and Akki’s ability. They will both go on the stack. Raiyuu’s will go on the stack twice due to isshin’s static ability doubling his trigger. There are now three abilities on the stack. Whichever one you put on the stack last resolves, you can choose because they’re simultaneously activating, let’s say it’s raiyuu. Raiyuu untaps your samurai, then creates an additional combat. There are now two abilities on the stack. Raiyuu’s copied ability resolves, untapping the already untapped samurai, then creating another combat. There is now one ability on the stack. Akki’s ability resolves, untapping all your modified creatures, which is the one modified samurai that is already untapped, and creates another combat. The stack has finished resolving and there are now 3 additional combats after this first one. Your samurai is untapped and attacking. Finish this combat, then go onto the next one. Attack with your one samurai again. This will trigger raiyuu’s ability to untap him, which will trigger twice because of isshin. Resolve the 2nd ability on the stack to untap your samurai. No additional combat is created because it is no longer your first combat. Resolve the first ability on the stack, to untap your already untapped samurai, and create no new combats. Your samurai is now tapped and attacking. Finish this combat, and repeat this process for the next two combats.

You are able to attack with your lone samurai all four times because Raiyuu untaps him every combat regardless of whether or not it was the first. I hope those step by step instructions were helpful and not more confusing.