Without a pump - which is either expensive or requires an electrical outlet - you are not going to get enough pressure to use a sprayer. You could put that barrel 10' in the air and it still wouldn't be enough gravity presssure.

I ran into this same problem when setting up an irrigation system. . Nearly tore my hair out. Bought a gravity-feed soaker hose, which was absolutely useless. Didn't want to have to install a pump.

So I bought 1/2" black plastic irrigation supply hose and the gadget you use to punch holes for feeder lines, snaked the hose at honestly a pretty low angle, and then just punched holes on the underside of the hose wherever my plants were situated.

Here's photos from before I put down mulch: https://imgur.com/a/B0CA4f9

From the looks of what you bought it appears the 1/2" supply hose might friction-fit right on that. If not, you'll have to replace that spigot with a regular hose bibb (which should just screw in) and then get the adapter to screw to that.

NOTE: IME you can NOT use Ts, Ls, etc. in the line; it has to be one long snaked piece. BUT obviously you are going to need to cap the end!

Hose bibb: https://www.amazon.com/gp/product/B0013ST3LU

Adapter: https://www.amazon.com/gp/product/B0049C774W

Supply line: https://www.amazon.com/gp/product/B00J9Q2TC8

Hole punch: https://www.amazon.com/gp/product/B004LOIMHE