Congratulations Pete on getting your first bit of work!
I would say that this probably a
good way to start, as there won't be too much ladder work to worry about, which is often the most tricky thing.
I don't want to make myself look a wally.
I don't think there's much chance of that, I've seen window cleaners doing shops, using only a scrim, and making a complete mess of it. Just the fact that you have the correct equipment will make you look like a pro.
Take your time, if your up a little ladder, do it in sections (even though its only one pane of glass).
Eg Top left, mop,squeegee, scrim
>> move ladder to middle,
mop(overlapping what you just did) squeegee, scrim
>> move ladder to right,
mop, squeegee, scrim.
Then do the bottom half at floor level all at once (overlapping what you did at the top), with long horizontal strokes.
You'll need your applicator(mop) quite wet, and after squeegeeing, you'll need to scrim the overlapped bits to dry off the excess water.
Hope it goes ok! Don't apologise for taking a long time, that just shows the owner you're willing to do it properly.
Philip