Fuchsia may last through the summer in your zone.

In this type of relatively shallow container, care is particularly important to how well the plants do. Plan on keeping an eye on the watering, especially in hot weather, and use a dilute fertilizer in the water or slow release fertilizer pellets. I also use a small amount of "water crystals" in my containers since they help reduce the frequency that watering is needed to once every day or two in the hottest weather.

How much sun is partly shaded? Do you want only one kind of plant or will several be OK? Many folks like at least some taller plants to go with the plants that spill over the side.

If you have 5 or 6 hours of sun, most full sun plants like the trailing types of petunia will do OK, especially if the shade is bright.

If you have mostly shade, look at fuschias, some of the more trailing forms of begonias (these are actually tender perennials and can be over wintered in the house if you want to carry them over), and trailing forms of Torenia.

Also look at plants which may flower some, but have colored or variegated foliage. Lamium (which also goes by the unattractive name of dead nettle, and is a hardy perennial) will have flowers for some of the time and variegated foliage all season. There is a type of Tradescantia AKA Wandering Jew that has deep purple foliage and is often used as a house plant in colder climates that is fine in a shady outdoor pot. Variegated Plectranthus (green and white leaves) will get huge and does fine in part shade as does sweet potato vine (chartreuse, maroon, or pink,green & white variegate). Lysimachia punctata AKA has small bright lime leaves and there are a variety of variegated forms of ivy. Both of these are perennials that have the potential to be invasive, so don't let them escape.

I would suggest going to a local greenhouse with a good selection of annuals and checking out what they have available. Read the tags as to water and sun needs as well as growth habit and see what appeals to you. That is generally how I choose my potted annuals each year.

What about drainage? Is there a hole in the barrel lid? If so won't the water from the pot go into the rain barrel carrying dirt with it? If there isn't a hole the planting medium won't be able to drain. How about setting another container into the hollow so it can have drainage but not pollute the rain barrel?

Nasturtiums might be nice. They are easy to grow from seed.

Lobelia erinus
even alyssum might work

As mentioned, depends on just HOW much part sun you're talking as well as how much rain you typically get, and how hot that area gets.