how conflicts between ministers and prime ministers from coalitions are resolved?

Usually by ending the coalition (or threatening the move). This leads to expedited elections, which happens a lot in Austria, although usually due to different circumstances like corruption scandals. Alternatively, a different government can be formed, but that is usually very unlikely (though it happened in 2019 when the president appointed a party-neutral interim government that was accepted by the parliament).

The problem Nehammer (the current head of government) is now facing is that regular elections are coming up in September anyways, so it doesn’t really matter whether he quits the coalition. Quitting is actually a bad idea, because then he wouldn’t go into the election as the government party, which can lead to a worse outcome.