To solve this problem, we follow the instructions given for moving the point $P(-2,-3)$:

1. Moving $1$ unit to the left means decreasing the x-coordinate by $1$. Therefore, the new x-coordinate is calculated as:
$$-2 - 1 = -3.$$

2. Moving $3$ units up means increasing the y-coordinate by $3$. Therefore, the new y-coordinate is calculated as:
$$-3 + 3 = 0.$$

Combining these two steps, we find that the coordinates of the resulting point after these movements are $(-3,0)$.

Therefore, the correct answer is $\boxed{\text{A}}$.