arena unlike pvp just counts on total initative, whichever team's total initative is higher, all their units will move first (by position of course)

and how do I know what my total initiative is

Go to Character, then Details and manually add the initiative for all 4 of your ninjas in your lineup.

No, arena considers total initiative only for the barrier, like outside it, but for the single moves does considers the single initiative you own in the current prompt move you are using.