Hello and welcome! I am new to this board but have been on a horse for 59 of my 61 years - lol
Yours is a very good question.
Some show horses are extremely adaptable to other venues such as trail riding. Some are not. It is not limited to a specific breed or mix breed.
I once looked at a gorgeous Morgan show mare who was terrified of the rustling of corn stalks as I tried to ride her down the tractor path at her owner's barn.
I once rescued a 5-gaited Saddlebred who was 17H and as big a baby as his 17H stature. He hyperventilated on the trails so I had to find him a "show home".
Then I bought a Morab (Morgan/Arab) who had gone ring sour so the owners wanted to sell her. From the second her front hooves hit the trail she thought she'd gone to heaven. She was a fantastic trail horse with very little coaching from me.
So regarding this Arab's ability to "switch gears", I would ask the current owners to ride her in the alternate environment you plan on using her in to see how she does. And with you watching
Also of importance is continuing to use the same bit as long as the horse is happy and performs well with it.
Saddle fit is important too. ESPECIALLY with Arabs as they are short backed and can sometimes have wide withers. My old Arab doesn't, but I have seen some that do.
Regarding feed. If the horse is doing well and is healthy, ask the current owners what and how much grain and hay they are feeding and feed the same.
The difference in amounts will vary according to how much pasture the horse will have once it comes to live with you.
LESS grain is always better unless the horse is going to be heavily worked every day.
I am dead set against sweet feeds of any type. So if this horse is being fed sweet feed, that would be the exception to my previous comment. Find something else
I hope this helps you some:)