They are both armed with their standard weapons, Rinzler has his two Identity Discs and V has two of his knives, and they are, as you can see above, already fighting. They are fighting in the Arena, so there will be no outside interference, and if Rinzler finds out and then questions the fact that V is technically a User he will be told to kill him anyway. V's knives work the same way Identity Discs do, as weapons, in this scenario.
I'm aware the knives wouldn't necessarily work the same way the Discs do, but that's a pretty stupid thing to bring up, considering what this site is for.
Fair point about Rinzler, though, he was very agile and acrobatic, but you're somewhat underestimating V.
No, it's not a one-hit kill in the Arena. If you're a Program you only die if you get hit in the torso, it's known as being Derezzed, but Programs can lose limbs, and presumably get cut, as well, and survive. The same thing applies to humans, or Users, as they're known on The Grid.
Also, V is easily agile enough to dodge Rinzler's attacks, and skilled enough to take him on in close-combat, seeing as he's somewhat superhuman and an incredibly well-trained, experienced and tough combatant.