Don't fertilize it until you start to see new growth on it and then just at half strength. After the first bloom is done you can give another dose. Don't feel bad about the black spot. The conditions at most nurseries tends to promote it. They're in a hot, damp greenhouse all crowded together. It happens. Keep it well watered and give it time to settle in and grow some roots and it should be fine. You don't say where you are but if it's very hot there you might want to shade it for a while until it's growing. Put it some place where it will only get morning sun but will be shaded in the late afternoon when it gets hot. Once it's growing you can move it out into full sun if you want.

The potting mix label should tell you whether it contains fertilizer or other nutrients. For example, MiracleGro says "feeds for up to six months." Also when roses are put in the nursery pot, they usually add some Osmocote (little plastic beads, maybe yellowish). For these reasons, it's good to be cautious. However, when I plant in the garden, I usually fertilize lightly as soon as the plant seems settled or has leafed out.

I'm in North San Diego, just on the edge of coastal/inland. Which means it'll get really hot this summer, but we've been experiencing what we call "May Gray/June Gloom". So it's been overcast and cool-ish over the last month. The sun is trying hard to come out, and might peek out this afternoon, but we're not seeing more than 4 hours of sunshine in a day. Some days we don't see the sun at all. It's possible we'll continue with this weather pattern into early July. I'll take this over last May though -- super dry, very high heat (90s++), and wildfires.

Regarding the soil I used, it's called EB stone Rose Grow Planting Mix. Doesn't say that it has fertilizer, and the ingredients say it's made from fir bark, chicken manure, peat moss, earthworm castings, pumice, bat guano, and that's all I can remember.

Chicken and bat manure are fertilizers good for a few months.

Since your not getting much sun right now anyway just leave it out to get as much as it can. IF it starts to get very hot and sunny and the rose is still struggling then you might want to give it shade in the hottest part of the day. If it looks healthy and growing well don't worry about it. Just leave it in the sun and keep it watered. That's a good healthy mix and your rose should be very happy with it without adding more fertilizer for some time. Disneyland is a really beautiful rose so sit back and enjoy it!