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Jax is really strong in a 1v1 situation. Tryndamere and Olaf are also good 1v1 champs. Poppy's whole kit is designed to focus a single champ and beat the **** out of them, so she's pretty strong in 1v1. Pantheon is one of the strongest in my opinion. A good Pantheon can beat pretty much anyone in a 1v1 early - mid game. His late game falls off a little, but he can still trade blows with the best of them. Malzahar and Kassadin are probably the two best 1v1 AP casters in the game. Warwick's passive lifesteal and natural tankiness makes him a really strong 1v1 champion, and the same goes for Nasus, especially if you can farm your q. Sion can be pretty scary in a 1v1.