I am looking for possible numerical methods to solve the PDE $$u_t+c u_x= \frac{-c}{x}u$$ for $u(x,t):\mathbb{R}\times \mathbb{R} \to \mathbb{R}$ and where $c>0$ is a constant.
I am particularly interested in a Finite elements method, although I am also curious if you can expose some other methods.
(I am refraining myself from writing the one or two methods that might be working to not influence the answers)
I am giving you alternative ideas to solve the PDE numerically.
Try using Fourier transform: let $\hat{u}(\omega,t)$ be the Fourier transform of $u(x,t)$. So using the inverse Fourier transform, we can write $u$ as $$ u(x,t) = \int_{-\infty}^{\infty} \hat{u(\omega, t)} e^{2 \pi \omega x (i)} d \omega $$
and
$$ u_{t} =\int_{-\infty}^{\infty} \hat{u}_{t}(\omega, t) e^{2 \pi \omega x (i)} d \omega $$
$$ u_{x} =-(2 \pi \omega)^{2} \int_{-\infty}^{\infty} \hat{u}(\omega, t) e^{2 \pi \omega x (i) } d \omega $$
$$u_{t} + cu_{x} = \int_{-\infty}^{\infty} \hat{u}_{t}(\omega, t) e^{2 \pi \omega x (i)} d \omega -c(2 \pi \omega)^{2} \int_{-\infty}^{\infty} \hat{u}(\omega, t) e^{2 \pi \omega x (i) } d \omega $$
$$ = \int_{-\infty}^{\infty}[ \hat{u}_{t}(\omega, t) -c(2 \pi \omega)^{2} \hat{u}(\omega,t) ] e^{2 \pi \omega x (i) } d \omega $$
but your right hand side has $1/x$ which is a problem. If it does not have, assume it is just $c u(x,t)$, then we have
$$ -c u(x,t) = -c \int_{-\infty}^{\infty} \hat{u}(\omega,t) e^{2 \pi \omega x (i) } d \omega $$
So your PDE is equivalent with the below ODE (in frequency domain):
$$ \hat{u}_{t}(\omega, t) -c(2 \pi \omega)^{2} \hat{u}(\omega,t) = -c \hat{u}(\omega,t) $$
You can then solve this either numerically using finite difference, or solving it analytically. But to deal with your right hand side that includes $1/x$, you may need to do some adjustment.