Are these equations correct? (transfer function)
It's difficult to say, given the way you formulated your question, but since you are given the corner frequency \$\omega_c=500\$, then you know that the transfer function -- which is that of a highpass -- has \$\frac{1}{\sqrt{2}}\$ of the maximum magnitude, so then \$H(j\infty)=50.1\sqrt{2}\approx70.8521\$. This is the ratio of the two resistors. Now you have two equations of two unknowns:

$$\frac{R_2}{R_1}=70.8521$$
$$\frac{R_2}{\sqrt{R_1^2+\frac{1}{(500*1\mu)^2}}}=50.1$$

Which gives \$R_1\approx2\text{k}\Omega\$ and \$R_2\approx141.7042\text{k}\Omega\$. Here's a quick check with LTspice:



The voltage is taken after the second, inverting E source (for positive output).