Depends on what you mean by a "decent mathematics graduate program", and depends on what you intend to study in graduate school.
The GRE Mathematics Subject Test focuses highly on "calculus and its applications". The pure topics (abstract algebra, topology, set theory, differential geometry, abstract analysis, etc.) tend to be less emphasised. And the format of the exam (being multiple choice) means questions tend to lean toward computational ones in nature, and less so conceptual ones. (Theoretically a firm grasp of the concrete concepts should allow you to do all the computations; in practice it helps to have good computational abilities since there are lots of questions on that exam.)
This is to say that (a fact that I think most admission committees will recognize to an extent) while a bad GRE math subject score may raise a red flag, a good GRE math subject score cannot be equated with either having necessary advanced backgrounds or high mathematical maturity. So just having a good GRE score is, in most cases, not sufficient to guarantee you a place in a math graduate studies program.
However, neither does having an undergraduate mathematics degree.
How much the GRE score can substitute for an undergraduate degree depends from place to place, but I doubt either is used as the final adjudicator for admission. You also should want to have strong recommendation letters as well as good personal statements.
In your situation, perhaps the best advice you can get is by writing directly to the directors of graduate studies (and also professors with whom you'd like to work on your degree) to seek their opinion. Since you are in a track less taken, it may benefit your eventual application by calling their attention to this fact.
To complement the other answer and comments: the utility of ideas of category theory is not at all limited to "algebra", either. For example, the product topology on an infinite product of topological spaces had always struck me as disappointingly weak, and I wondered why that was "the definition". In fact, that construction is a construction of (a model for) the categorical product in the category of topological spaces. That is, instead of just being a definition we've inherited, it has functional properties. In fact, with this bit of hindsight, it seems to me perverse to "define" so many things without explaining what is supposed to be happening. A categorical characterization is often much more informative.
Another example, algebraic, but not "fancy": what is an "indeterminate", after all? A "variable"? Certainly there are heuristics that we'd tell beginners, and we know how to use "indeterminates x,y...", but what are they? One precise form is to say that $\mathbb Z[x]$ is the free ring with identity on one generator $x$... meaning that, given $r\in R$ in an arbitrary ring $R$ with identity, there is a unique ring homomorhism $\mathbb Z[x]$ to $R$ sending $x$ to $r$.
In a quite different direction, the topology on the space of test functions on $\mathbb R^n$, or even just on compactly-supported continuous functions, is a colimit. For continuous compactly-supported, it is the colimit of Banach spaces $C^o_K$ of continuous functions on $\mathbb R^n$ supported on compact $K$. In contrast, the "definition" given in Rudin's "Functional Analysis" for the test function topology is actually a construction, and the following section proves several mysterious lemmas which, I only realized later, were, in effect, verification of the colimit properties. (Indeed, Schwartz overtly used the notion of colimit c. 1950, but use of such notions had been out-of-style for U.S. analysts for many decades. Some of that may be anti-Bourbaki reaction, even though Bourbaki did not use category-theory ideas, either.)
In the short term, it is usually possible to "get along" without overt use of category-theory terms or ideas. However, the more things one finds reason to remember, the more imperative there is to organize them well, to eliminate redundancies and duplications and waste, etc. Category theoretic ideas are very helpful in this regard.
(This is not to say that "formal" category theory is necessarily as broadly useful, in the same way that, while set theory is undeniably useful, the utility of a highly developed formal or axiomatic set theory is probably not nearly as useful as the basic parts.)
Best Answer
Perhaps the most compelling answer to the question:
is just that a great deal of modern algebraic geometry, as a result of Grothendieck and others, makes heavy use of category theory. Here, I'm basically replacing "structured sets" with "rings, fields and modules." However, the study of groups and monoids has also been immensely impacted by category theory. In particular, category theory has played a really large role in representation theory.
If "structured sets" also includes topological spaces then there's even more category theory that ends up being relevant. And of course, topological spaces, and the study of their invariants, are important in things mentioned above like algebraic geometry and representation theory.
One problem with your question is that it's immensely broad. You're basically asking "How is category theory used in algebra?" And well, the answer at this point is almost "In what cases is it not used?"
My suggestion, if you want to have some sense of what category theory is all about, is pick up Saunders Mac Lane's book and force yourself to learn the foundations (e.g. categories, functors, natural transformations) from a purely formal point of view, and then read about the examples. Then pick some topic you like (e.g. ring theory, or representation theory) and ask a more specific question about category theory in, say, representation theory. Again, for any of this to make any sense at all, you'll have to have a pretty good grip on whatever topic it is you're interested in. Category theory tends to produce "large scale" structural theorems, and so if you're not familiar enough with a topic to be interested in how all of its pieces fit together, it (in my opinion) will be very hard to motivate category theory.
However, after writing all of this, and then reading through the comments above a bit more, I see that someone has really already provided you with an answer, which is this MSE question, so that's probably a pretty good place to start.