Udyr is a strong dueller in the early game. Unless you have a clear advantage, avoid fighting him in a 1v1 scenario.
Udyr has good objective control and can solo objectives like the Dragon with ease. Make sure you keep the Drake warded at all times to decrease his chances of securing them for free.
Unlike most champions, he doesn’t peak at level 6 as he can unlock his Ultimate R at level 1. If you have a strong Ultimate, try to abuse the cooldown and help your allies.
Udyr has no reliable engage tool other than running towards your team. Kite him so he is unable to land his CC onto your carries.
If he ever uses his Flash E to engage, remember that he has no other escape tool. Focus him down as quickly as possible.
As he isn’t really good in team fights, you could force a team fight while he is in his jungle farming, split pushing in a side lane or otherwise preoccupied. Fighting with the numbers advantage will make team fighting much easier.
Udyr is a good early game Jungler. Avoid giving him kills to reduce his ability to carry in the mid-game.
Trinity Force is a key item on Udyr. Once he has this item, taking him down alone will be difficult. Team up with a buddy or two to make killing him easier.
Udyr falls of in the late game, but he can still be useful thanks to his E. Avoid overextending if you know Udyr is nearby.