Am I on the right path to solve this RLC-Circuit? (Need to find the voltage at the resistor)
Yes, your analysis is correct.

It would have been quicker to solve for the resistor current and then multiply through by \$\small R\$ to get the resistor voltage, \$v_R\$, as below (I've used \$\small j=\sqrt{-1}\$ to avoid confusion with current, \$i)\$.

Let \$i\$ be the resistor current, then the voltage across the current source is:
$$L\frac{d}{dt}(I_0e^{j\omega t}-i)=\frac{1}{C}\int i\:dt+Ri$$
differentiate,

$$L\frac{d^2i}{dt^2}+R\frac{di}{dt}+\frac{1}{C}i=L\frac{d^2}{dt^2}(I_0e^{j\omega t})$$

multiply by \$\frac{R}{L}\$,

$$\frac{d^2v_R }{dt^2}+\frac{R}{L}\frac{dv_R}{dt}+\frac{1}{LC}v_R=-R\omega^2I_0e^{j\omega t}$$