Would the bacterium developing from staphylococcus aureus into methicillin-resistant Staphylococcus aureus an example of anagenesis?

It's evolution. It doesn't happen to a single bacterium. Evolution doesn't happen to individuals but to populations.

The development of bacteria resistant to antibiotics was inevitable. When a person is infected and they are given antibiotics some baceteria may survive because they have a mutation giving them resistance to the antibiotic. They become the founders of a new population. The oversue of antibiotics has help resistant strains develop. By continuing to overuse antibiotics the environmental conditions are created for the resistant bacteria to survive. Bacteria with the mutation increase in frequency. Ultimately you end up with populations where every individual has the resistance mutation.

An individual bacterium can't change from S. aureus to MRSA. What changes is the population of bacteria. Individual bacteria do not undergo changes in their lifetime. What happens is when a colony of bacteria is exposed to a ineffective dose of an antibiotic, some of the colony are killed, leaving only the ones that are antibiotic resistant due to some genetic mutation. Those individual reproduce, meaning that subsequent doses are going to be less, and less effective, as more and more of the population are MRSA.

It's a good example of how natural selection drives the evolution of a population. Anagenesis is a quicker version of evolution.