EDIT: As Angelo mentions, the argument below has a problem in the case of non-reduced curves. I am not sure that an upper bound for the arithmetic genus is impossible to show, but certainly the lower bound for Cohen-Macaulay curves is false, as Angelo's examples show. The argument below shows that there is an upper bound (and in fact there is also a lower bound) for the arithmetic genus of a reduced subscheme of pure dimension one in projective space, whether Cohen-Macaulay or not. I tried to play a little with the Cohen-Macaulay condition to prove that there is an upper bound, but with little success.
Choose an embedding of X in projective space. Since your curves are all in the same homology class, they all have the same degree: this is simply the intersection number of the ample class with the homology class of the curves. Generic projection to a plane tells you (since the curves are reduced) that the curves you are interested in are partial normalizations of plane curves with bounded degree. Since the arithmetic genus decreases under (partial) normalizations, and since the arithmetic genus of a plane curve of bounded degree is bounded above, you conclude that the arithmetic genera of your curves are bounded above.