Hey Plague,
that is a great question considering both specs are soo far apart it feels almost like playing a different class to some extent.
now regarding arena, arms is indeed “harder” to play specially (as you pointed out) due to the lack of self-heal which. Since you play mainly Fury this will be quite a head turner for you and my main advise is to practice practice practice, but practice what? you ask.
For starters you go into BGs, you will see soon why i don’t suggest arenas quite just yet, as arms your defensive CDs are the hardest to manage, because again no self-heal, therefore you need to use these only on the most crucial points and if you mistime it or save it for a even more clutch situation, you lose.
every arms warrior as been through this and you only learn by practising, better in BGs than arenas since its a bit more chill and depending on the arena you can pick your targets according to your comfort level as you go on this path.
you will fail a lot, and you might get upset. But remember, arms warrior is NOT a solo spec for pvp, the sooner you understand this the better, its dependent on his team to go full 
ham , and once it goes its literally a juggernaut if played correctly.
with all this said, you can still play Fury spec, but mainly in 2vs2. Fury bring little utility compared to arms and that’s why it can play a role in 2s, regarding 3vs3 then you have to go arms since its an amazing team player.
little summary regarding def cds:
Avatar (azerite trait Bastion of Might 
very important 
- this is the def cd you use the most deliberately, don’t save Avatar just to increase your dps, use it for defence as well, because when you do means you can go 
all in a bit more, again you gotta have the azerite trait (to unlock it you have to spec protection btw)
Rallying Shout - this is the cd you use on the majority of the time (varies with experience and situation) BEFORE Die by the Sword
Die by the Sword - is basically the warrior “bubble”, this cd if used incorrectly either too soon or too late or even when you didn’t need it at all and your healer had you all along will make you lose the arena, period. (don’t use it when you have like 7% hp, specially vs casters)
all this comes with practice and a lot of 
feelsbatman moments but the rewards are nice in my opinion.
its a slower pace, comparing to fury, but every hit really does matter.
i hope this has managed to help you somehow Plague and feel free to keep asking, i am sure the rest of the warrior community will be glad to help you out.
oh and remember to check some guides on youtube, helps a lot.