Are you morally opposed to installing a gutter on the shed? Probably not; perhaps it is a question of saving money. Then the downspout can collect from the gutter and into the barrel as designed.

If it just for a shed, you might find serviceable gutters left at the side of the road as destroyed houses get fixed or demolished.

If you live in the north, it can freeze in the winter and you are out of luck unless you bring buckets in the warm house before the freeze. Also the barrel itself might not be frost hardy- a freeze with water in it could split the seam. Certainly a tap at the bottom of the barrel would freeze.
But to answer your question about directing the water, I once used plastic sheeting to create a water flow out a window from my leaking enclosed porch roof. Something like a plastic shower curtain might also work

We do have freezes but they don't last--we can use regular buckets at those times.

The plastic shower curtain idea is a gem--we don't need a big system so buying a tarp would be overkill. Thanks for that!

I used a shower curtain too and the ring openings are handy to hook over nails or whatever. You might have to secure it in a windy situation though! Best of luck!

is to build up a trough along the side of the shed. Build it up over the size of the barrel, and then have it funneled down into the barrel. Like I said, probably goofy... but I feel ya. Good luck!

How long is the drip side of your shed, and how far from the shed wall does the drip edge overhang? If the drip edge is about 2-3 inches from the shed wall, you could just screw a regular piece of rain gutter to the wall of the shed just under the drip edge with a cap on one end and a down spout on the other. If you attach it to the wall, it won't matter if the roof profile is flat.

Good luck!

Almost the whole side of the shed is protected by the roof from rain.

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You might be able to make something like this work. Looks like you could bolt it directly under the roof using nuts, bolts, and plastic washers right near the bend of the flat edge of the gutter so it wouldn't bend with the weight of the water. It's called a "half round highback gutter." Sorry, I don't have any other ideas. Good luck!

We don't need a gutter to run all the way across the bottom of the roof, just enough to collect some water over time. Thanks!

Put up a tarp?
https://harvestingaqua.com/collect-rainwater-without-gutters/


Some states ban it.
https://www.familyhandyman.com/article/collecting-rain-water-in-these-states-could-be-illegal/

The gutter someone else suggested turns out to be way more expensive than we need. This tarp idea, made on a much smaller scale than at the website, seems like the best bet yet. Thanks for the idea, I appreciate it!