I have Irish citizenship through my grandmother and an Irish passport.
You need to get the citizenship first. Its pretty easy as long as you have the proper records. Once you get that, its as easy as paying a fee and taking a picture to get the passport. If I remember correctly the entire process took about 6 months.
We had a chat with a guy who went through the process for dual citizenship. It took a bit of effort to get the papers in order, but now he's thrilled to use it. You can have a listen at