If your autosomal DNA match is strongly consistent with a first-cousin relationship (see
https://isogg.org/wiki/Autosomal_DNA_statistics for statistics to compare with), you can be confident that you really are biological first cousins.
And after that, as Peter says, when you compare your data and your cousin's data with new DNA relatives, it can help indicate which side of the family the new relative comes from (assuming you and your cousin are related on only one side of your family). If both you and your first cousin have good matches with a new DNA relative, it's a good indication that the new relative is on that side of the family.
Because this cousin is a male and his mother is your father's sister, you will not share yDNA, mitochondrial DNA, or X-DNA (except possibly by coincidence). X-DNA is part of the autosomal DNA test. If you should find a DNA relative who shares X-DNA with him, that cousin is assumed to be on his mother's side (your father's side of the family). If you find a DNA relative who shares X-DNA with you, that relative is on your mother's side.
But none of this matching is "automatic" -- you need to look at the data and the genealogy. WikiTree DNA tools are designed to help with the data analysis.