I would like to use this plasmid (mammalian expression Flag-HA-USP53 (Plasmid #22606) - Addgene) to produce plasmid in e coli to purify plasmid. Would this work? I am starting out with this vector because it is already available and has the insert I need.
Short answer: Looking at the plasmid map, yes, this should work.
Long answer: It depends on the plasmid. To be able to replicate a plasmid in bacteria, you need a origin of replication (ori). To find out if this is present, have a look at the vector map provided by Addgene:
It shows the ori on the left side of the map. Additionally an antibiotic resistance gene is helpful to be able to select for bacterial cells whic have taken up the plasmid in the transformation, which is also present in form of the ampicillin resistance gene.
Generally speaking most expression plasmids will carry both feature since bacterial transformation, growth in culture and mini (or midi or maxi) prep it is by far the fastest, easiest and cheapest method to produce large amounts of plasmid.