This is a simple (yet not simple to solve!) transient heat transfer problem using cylindrical coordinates. There are solutions available in handbooks if you do not want to go through the math and Bessel function solutions arise. Its been a while since I have done it, but I found this myself.
The The boundary condition is the tricky part. If you are using convective boundaryboundary condition with known $h$, then you can use these charts provided by Cengel. With radiative heat tranfer, its a bit tricky. You would need to use Stephan-Boltzmann law and somehow estimate the emmisivity of your surface, as well as the view factor. is the tricky part. If you use a constant temperature for the surface, i.e, Dirichlet BC, the problem is considerably easier but would not represent the reality of the situation. In reality, its is convection and radiation that is responsible for the heat transfer in the exposed areas and conduction where the cylinder rest (i would assume the base from the way you have drawn it). If you are using convective boundary condition with known $h$, then you can use these charts and how to use them provided by Cengel (also attached herein).
With radiative heat transfer, its more complicated. You would need to use Stephan-Boltzmann law and somehow estimate the emmisivity of your surface, as well as the view factor. However, you could in come up with an conservative heat transfer coefficient, $h$, that could allow include the heat radiation part as well.
