Will this strategy work for verifying successful cloning of a DNA fragment into a plasmid vector?
Yes, your strategy would work, with the caveats already mentioned by @Cell.
Depending on which restriction enzyme and PCR buffer you use, you might even be able to skip step 2. NEB has a good list over their restriction enzymes and their activities in some common PCR buffers. With some reasoning that information is often applicable also to restriction enzymes of other brands.
Depending on your primer availability, another quick strategy is to simply use a primer pair such that one primer binds inside your fragment, and the other binds to the backbone. This should result in a PCR product only if your cloning was successful, and also verifies correct direction (in case you used blunt-end or single-site cloning). Two possible problems with this approach is:

If you get no bands at all you may not know if what failed was your cloning or the PCR reaction. The latter can be easily controlled for with a well-thought positive PCR control.
You may detect cases in which more than one copy of your fragment have been inserted. Although such cases are rare, it sometimes happens that inverted repeats of 3, 5, 7, ... etc copies get inserted in tandem.