May in Edinburgh is renowned for its ephemeral weather. It might be sunny, yet chilly; it will be overcast, but you will sweat when wearing a jacket. The winds might be brutal, making all that hair product go to waste.
Arthur's Seat is a highest point of the Holyrood Park. Even though it is a stone throw from the Holyroodhouse Palace, I would not recommend combing visits to those two together. The visit to the palace will take a few hours and will probably drain the energy. I would suggest visiting the Arthur's Seat earlier in the morning or after having a light lunch after you still have energy for a hike.
There are many tributary trails, so for those who love to hike, will spend longer time exploring the park. There are also many view points and amazing flowers, so photographers will enjoy the outing.
For those who love history, there are the ruins of ak old church. Explore those!
The last part of the trail to the Arthur's Seat is a bit steep. If you are not an experienced hiker, stay on the main path, as the side routes are rocky, unmentioned, amd crumble under your feet. Also, wear athletic shoes (my addiction to Vans nearly ruined a new pair on the climb).