A few caveats: absolutely make sure that the outlet you are plugging into is a GFI. You should also have it fitted with and "in use cover".
Ensure that the extension cord that you are using is in impeccable condition. No cracks in the jacket or exposed insulation, and especially no exposed wire. Make sure that where you plug in the lights to the extension cord, you are off the ground. Possibly reserve this cord for this use only.
Fluorescent tend to not like to strike in the cold. They have improved in recent years, but I could not possibly tell you that what you have will strike at say 40 deg F.
Now I really should be trying to encourage you to do this the legitimate way. How hard is it to dig a trench? You need to go 18" down. That is not that deep. You almost certainly would get by with a 30amp 120/240 service. That is three lengths of 10gauge from a two pole 30a breaker in your main panel out to a sub panel in the shop.
Run it in plastic conduit. You could use 1/2" but spend a tiny bit more and use 3/4" and the wire will pull itself thru the conduit. To avoid needing to purchase a large fish tape, which for one job is a bit of an investment, pull some pull string (there is a specific product in the electrical dept) thru the conduit as you assemble it.
Square D has a great 6 space breaker panel that costs next to nothing that I use a lot on gigs like this. You can have a 15a breaker in there for all the lighting, and a 20a breaker for any tools, and still have room for a 240v tool like a welder or air compressor. (A 240v air compressor will deliver LOTS of air!)
You need to drive a ground rod near the new sub panel, and you need to ensure that any neutrals in your new sub only connect to the neutral bar in the sub, with no connection to the grounds or the case of the sub panel, and that all the grounds and metal components be bonded together and grounded to the ground rod. (You do not attempt to link the grounds in the shed to the ground in the main panel).
If there is any plumbing in the building, it should be bonded to the ground as well to ensure that it is at the same potential as the ground rod and the grounds running in the shed.
If the materials all come to $250 I'd be a monkey's uncle.