It is actually not a predicative adjective (adjectif attribut in French) as it is not linked to the subject through the main verb. It is a past participle used in a participial phrase.arrivé is a past participle (used as an adjective) describing a state of the subject (predicative adjective).
This implies that the noun it refers to, namely the subject (here: je), is masculine singular.Arrivé here seems to be masc. sing. but I am not sure why.
It has indeed nothing to do with une fois, which may actually be omitted here: Arrivé au bureau, j'ai remarqué que…I had thought arrivé was related to ‘une fois’ but apparently not.
I guess you could use exactly the same construction in English :Une fois arrivé au bureau, j’ai remarqué que le dos de ma chemise était sale
Which is translated as: Once I arrived to the office I noticed that the back of my shirt was dirty (not great English - should be at the office.)
As far as I know, in modern English it is better to repeat the subject. "Once arrived" sounds really weird, at least to these American ears.As far as I know, you should repeat the subject in English:
Once I arrived at the office, I noticed....
Une fois arrivé au bureau, j'ai remarqué...