Chonaic mé - - Literally: I saw her at the disco on the Saturday before that and I wanted to make a date with her.

Chonaic mé í - - I saw past tense of feic - - see (Do not use feic as a command use féach - - look at instead)

ar an Satharn - - roimhe - - sin - -
on the Saturday - - before it - - that
i.e. on the previous Saturday.
The word ar could be left out here and the words an Satharn roimhe sin would qualify chonaic mé adverbially.

Roimh - - before is one of the simple prepositions. It occurs in the familiar phrase céad míle fáilte romhat - - a hundred thousand welcomes before you i.e. you are very welcome.

Theastaigh uaim - - I wanted.

coinne a dhéanamh léi - - to make a date with her