Will Fst (inbreeding coefficient) always be a value close or at 0 in asexual populations?
It depends on whether mutation exists. Generally mutation is happening in biology, but in its absence, yes it will be at or near zero:

Note that in the absence of any mutation, $F_{ST}$ would be defined but equal to 0, as all the genetic variance is within individuals and none between individuals and subpopulations.

From that paper, see this figure:

So, basically, the exact level of $F_{ST}$ in such cases will be controlled by the mutation rate. That figure shows a value above zero at 100% clonal reproduction because it does have a nonzero mutation rate. There is also nonzero migration between populations, so that will have the effect of reducing $F_{ST}$ somewhat. So here the value is controlled by those 2 factors. Another thing worth noting is that, if there is no initial variation between populations before the split, that will tend to increase $F_{ST}$. So it does matter how much variation exists overall.