Answer: 803

Small Hint:

The garden is itself a hexagonal block of unit hexagons with $n - 1$ on each side; count it as a center hexagon plus rings of $6,$ $12,$ $18,$ $\ldots$

Big Hint:

The garden holds $1 + 3(n-2)(n-1)$ hexagons, and each unit hexagon is $6$ equilateral triangles of side $1$

Solution:

The garden enclosed by the path is itself a hexagonal arrangement of unit hexagons with $n - 1$ on each side. Counting from the center outward in rings of $6,$ $12,$ $\ldots$ hexagons, it contains $$\begin{aligned} &1 + 6 + 12 + \cdots + 6(n-2) \\ &= 1 + 3(n-2)(n-1) \end{aligned}$$ blocks, which for $n = 202$ is $1 + 3 \cdot 200 \cdot 201 = 120601.$

Each unit hexagon consists of $6$ equilateral triangles of side $1,$ so its area is $6 \cdot \frac{\sqrt{3}}{4} = 3 \cdot \frac{\sqrt{3}}{2}.$ The garden’s area is therefore $3 \cdot 120601 = 361803$ times $\frac{\sqrt{3}}{2},$ so $m = 361803,$ and the remainder upon division by $1000$ is $803.$