Just fill in a table
There are only eight possible out comes, so write vertically 1 and 2 for x
and 1,2,3,4 horizontally for y.
Now fill in the eight positions and go from there
Next, obtain the marginals by summing across and down.
The event of X+Y=4, occurs when X=2 and Y=2 OR X=1 and Y=3
Just add the probabilities of those possible outcomes.