While not as easy as the Flight of the Eagle Spirit, Arena of the Wolverine is still a rather forgiving activity, which you can see from the video guide embedded below.
Anyway, the activity is pretty much a DPS check so bring your best damage pumping build and slap the Wolverine with it. As long as you have a competent build, the creature shouldn't be able to stop you from getting gold but if you want to compete on the leaderboard or have low damage output, some of the following tips might be useful.
The first tip is to lure the spirit away from the entry point. This step is here to make sure the game doesn't prompt to you to exit the instance all the time.
Another important thing to keep in mind is the Wolverine's signature mechanic - it will absorb some spiritual energy and then shoot it out. If this hits you, it will cause a stun, slow and chill. To avoid this attack, it's best to remember it happens in regular 10-second intervals and the AoE stun is always preceded by an animation where three spirits are absorbed into Wolverine.
You can either use the regular dodge or abilities like the Warrior's Whirlwind Attack or Thief's Pistol Whip to evade the shockwaves. If you are using an evade that also changes your position, it would be wise to move towards Wolverine during the dodge. If you evade backwards, the slow-moving shockwave will eventually catch up and still hit you.
In case you are unable to dodge these shockwaves, it's a good idea to bring spammable condition removers. Use them to cleanse Slow and Chill caused by the shockwave.
The third tip is connected to Wolverine's ability that functions similarly to the Elementalist's Static Field. These are the clearly visible areas marked by blue circles so avoid their walls as they will also micro-stun you.
If you are just here for the gold medal, even if you don't adhere the aforementioned tips, you should be able to get the reward as long as you keep your DPS rotation going. Keep in mind that in the video I didn't have Berserker gear, back item, amulet, rings or accessories which dropped my damage output heavily. I didn't dodge the shockwaves either and still got to the reward with over 22 seconds to spare.
In other words - with enough damage output you can just brute force this activity's gold reward but if you are looking to climb the leaderboard, you need to adjust to Wolverine's mechanics.