From my understanding, this assignment wants me to calculate $q_{1}(t)$, $q_{2}(t)$, $p_{1}(t)$, $p_{2}(t)$ using Euler's method. I am a quite confused as to how to proceed.
For example, at $t_{0}$, $q_{1}(t)$ is $1-e$.
What about at $t_{1}$? $$q_{1}(0.0005) = (1-e) + 0.0005(0)$$ $$q_{2}(0.0005) = (0) + 0.0005(\sqrt{\frac{1+e}{1-e}})$$ $$p_{1}(0.0005) = (0) + 0.0005(\frac{-1}{((1-e)^2 + 0^2)^{3/2}})(1-e)$$ $$p_{2}(0.0005) = (\sqrt{\frac{1+e}{1-e}}) + 0.0005(\frac{-1}{((1-e)^2 + 0^2)^{3/2}})(0)$$
Are they correct?
Yes, this first step is correct. However, you will need to write some code, like for instance in this answer about Euler and RK4 in orbital mechanics, it would take far too much paper to do the $400,000$ steps manually.