Guest Tom,
"I walk alone" in Irish would be "Siúilim i m'aonar".

Siúl a rúin does indeed mean "Walk (imperative), oh sweetheart" (or other similar term of endearment)... the word "rún" (of which "rúin" is the vocative form because it is being addressed) literally means "secret". (Compare "secretary = rúnaí")

best wishes
D