As already mentioned 'Owen had his arresting' is how we use the passive tense in Welsh, ie something that happens to the subject of the sentence.
In English this is written 'Owen was arrested'
However a sentence 'Owen was arresting criminals yesterday' is not a passive one because it's Owen who's doing arresting. The difference is quite subtle in English, but much clearer in Welsh.
'Owen was arrested yesterday' = 'Gaeth Owen ei arestio ddoe.
'Owen was arresting criminals yesterday' = 'Roedd Owen yn arestio troseddwyr ddoe'
I had always thought cael was "get" (receive, obtain, acquire), which would have made this even closer to English, e.g. "I've got a car" (= I have a car) versus "I got arrested".
So things like ges i fy nhalu = I got my paying = I got paid.
Gweiadur only has "have" for cael, but geiriadur.net also has "receive, obtain; get".
Yes, that is an interesting spin on this.
Although the short past tense of cael eg 'ges i, gest ti' is universally translated as 'had'
eg "ges i frecwast" = I had breakfast "gest ti hufen iâ" = you had an ice-cream
and yes in some English dialects 'got' would also work here.
'Ap geiriaduron' (which is the best app dictionary by far) gives:-
"get/acquire/extract/have/obtain/procure" for "cael"