Nondeterministic PDA for the following language with Kleene star
I figured out the answer and hope this will help anyone with similar confusion in the future.

Since a pushdown automaton is essentially a finite automaton with an extra component called the stack which is used for storage, if we are presented with a regular language like the one given above, we simply need to create a PDA that does not use the stack. This concept confused me at first, but here's what I mean:

$$L = (aaa^*bab)$$

$\space$

$$M = (Q, \Sigma, \Gamma, \delta, q_0, z, F)$$
$$Q = \{q_0,\ q_1,\ q_2,\ q_3,\ q_4,\ q_5\}$$
$$\Sigma = \{a,\ b\}$$
$$\Gamma = \{$\}$$
$$z = \$$$
$$F = \{q_5\}$$

$\space$

$$\delta(q_0, a, $) = \{(q_1,\ $)\}$$
$$\delta(q_1, a, $) = \{(q_2,\ $)\}$$
$$\delta(q_2, a, $) = \{(q_2,\ $)\}$$
$$\delta(q_2, b, $) = \{(q_3,\ $)\}$$
$$\delta(q_3, a, $) = \{(q_4,\ $)\}$$
$$\delta(q_4, b, $) = \{(q_5,\ $)\}$$

$\space$

I'm not completely certain about the stack start symbol, because I've noticed some resources use it regardless, while others assume that at the beginning the stack is empty ($\lambda$) and push the start symbol at the first step.

Anyway, I hope this answer helps anyone else.