How to calculate or should I measure impedance for 1/4 Wave monopole?
A Vector Network Analyzer is the instrument normally used for this task. The VNA will give you one port s-parameters (S11) that you can use to synthesize a matching network.

However, if your ground plane is significantly "large enough" you can use an analytical approach and assume a feed impedance of 1/2 that of an equivalent dipole antenna.

Simulation accuracy varies from bad to very good depending on how well you understand the problem and the limitations of your simulation method. Like any FEA simulation, you still need to have a good intuition for the problem to get the most out of it. Most EM sim packages (HFSS, CST Microwave Studio, FEKO) are quite expensive, so I would suggest avoiding them unless you have very specific requirements that can't be met by prototyping.

That said, FEKO has a mesh size limited free version that would probably serve you well for a basic sanity check sim - monopole above a finite ground plane simulation.