"Tháinig mé ar" I find this makes more sense in translation if I think of it as "came upon"
The idiomatic translation of that is 'oidhreacht a fháil' = to get an inheritance. 'tháinig mé ar' would not sound right, I'd think. More likely, 'tháinig mé in airgead', 'tháinig mé isteach ar an airgead', 'thit airgead liom', 'thit airgead orm'.
More likely, 'tháinig mé in airgead', 'tháinig mé isteach ar an airgead', 'thit airgead liom', 'thit airgead orm'.
You can't say any of these unfortunately. They are variations of the English idiom.
oidhreacht a fháil
... is probably the best.
In the Newfoundland dialect, "come across" is much more common than "come upon"
Don't confuse what you are learning here with Newfoundland ;)
Tháinig mé transa ar x / I came across x
... is used in Irish also.
If we were to translate it more literally, it'd be like the English "I came upon the money.". It's what's known as a phrasal verb, and the ar is required with tháinig to give it the meaning of 'found'.
It doesn't work because it suggests that you looked the answer up in the dictionary. In modern Irish, when people say airgead, they're talking about money, not a shiny metal.
If you're learning Irish for the purpose of communicating with other speakers, assume that airgead="money", unless the context is quite specifically talking about the metal/element. "No context" really doesn't justify using dictionary translations that will be misleading or confusing if you're trying to use what you learn to communicate with someone else.