Learn Finnish: Satuin mainitsemaan asian hänelle aivan ohimennen. - I happened to mention the matter to him quite in passing.

When sattua is used to mean 'to happen to do something', it requires the following verb to be in the 3rd infinitive illative, which is the -maan/-mään form. Therefore, 'I happened to mention' translates to satuin mainitsemaan.

Why is the object asian instead of the partitive asiaa?

The verb mainita (to mention) expresses a bounded, completed action here. Because the specific matter was fully mentioned, the object takes the total object form, ending in the genitive-accusative -n.

What is the grammar behind the word ohimennen?

It is an adverb made from ohi (past, by) and mennen. The word mennen is the 2nd infinitive instructive form of mennä (to go), which is used to describe the manner in which something happens. Literally meaning 'while going by', it perfectly matches the English phrase 'in passing'.

I previously learned sattua as 'to hurt'. How can I tell the meanings apart?

You can tell by the verb rection (the required surrounding sentence structure). If sattua means 'to happen to', it is paired with a verb in the -maan/-mään form. If it means 'to hurt', it usually appears with a location case, such as jalkaani sattuu (my foot hurts).