Grandfather being your father's father, I assume?
Yes, Y DNA is an option but not the first I'd consider.
Y DNA is male only - father to son through the generations. Doing this test can be really good if you're trying to get far back on the male line as Y DNA doesn't change much through the generations.
However, Y DNA tests are far less popular than autosomal DNA tests, which can be used by both women and men. By doing a Y DNA on your brother you are both limiting the pool of matches within other people who have done that test, AND ruling out all the women who have tested too. Yes you might get a match with men whose common ancestor is in the 16th century - but that's not going to help you trace a very recent ancestor, is it?
My advice would be : Get a standard Ancestry DNA kit test. Ancestry is the best choice as they have the largest database and also allow you to download your raw data and upload it to other sites such as Gedmatch. Either you take the test or your brother does - makes no odds. If your father is still alive, would he be up for it?
Look at your matches, try to rule out matches which you think are on your mother's side of the family and not your father's. Try to identify a common ancestor for you and your matches and work forward - if you're matching with second cousins that will be a great grandparent. Rule people in/out - so women children obviously can't father a child, neither can people too old/young etc.
There are lots of FB groups which can help you with this. Another thing to consider is testing someone else you know is descended from your grandfather - a cousin or second cousin. If they are matching with the same people as you are matching with, then you know you have a match on the right side of the family.