Is the algebraic number theory in Ireland and Rosen enough to study Washington's Cyclotomic Fields?

I think you'd be happier to have a full introduction to algebraic number theory, like S. Lang's, or equivalent. Then you'll be able to see many of the features of cyclotomic fields as special cases of what would happen more generally, rather than having those special cases appear as novelties.

That is, I think it is useful to see the basic features of cyclotomic fields appear as especially accessible examples of general algebraic number theory, rather than as first-encounter extensions of the number theory of $\mathbb Z$ and quadratic extensions, for example.

The preface to the first edition of Washington's book says:

The reader is assumed to have had at least one semester of algebraic number theory (though one of my students took such a course concurrently). In particular, the following terms should be familiar: Dedekind domain, class number, discriminant, units, ramification, local field. Occasionally one needs the fact that ramification can be computed locally. However, one who has a good background in algebra should be able to survive by talking to the local algebraic number theorist. I have not assumed class field theory; the basic facts are summarized in an appendix. For most of the book, one only needs the fact that the Galois group of the maximal unramified abelian extension is isomorphic to the ideal class group, and variants of this statement.

It seems you'll have to learn much more about algebraic number theory than covered by Ireland and Rosen.