Fast polynomial calculation over $\mathbb{Z}_{487}$
You can very efficiently compute $a(x_j)$ for each of those 243 values of $x_j$ by simply evaluating $a(\cdot)$ using Horner's rule.  The running time is 243 evaluations of $a(\cdot)$, and each evaluation requires 243 modular multiplications and 243 modular additions.  That's a total of $243^2 \approx 2^{16}$ modular multiplications.  Consequently, the total running time will be extremely efficient, even without trying to split this into three polynomials $a_0(x),a_1(x),a_2(x)$.

If you want a more efficient method, a better approach is to use a Discrete Fourier Transform.  You can use a DFT over $\mathbb{Z}_{487}$ to evaluate $a(x)$ at 256 different points in running time that's asymptotically $O(n \lg n)$, where here concretely $n = 256$.  [Alternatively, you could use the Number-theoretic transform for $\mathbb{Z}_{487}$ to evaluate $a(x)$ at $486$ points, and do all your calculations in $\mathbb{Z}_{487}$.]