For my current project I do a lot of research about aaRS but didn't find answers to these questions yet. I am happy about any information about the lifecycle of these enzymes.
proteins are usually stable up to 1 or 2 days in cells, RNA is far less stable and degrades within some minutes or up to some hours. Protein and RNS stability depends on the species. Higher species have much longer protein and RNA life time.
See the following webpage, it has some interesting figures