My sister has a Purple Passion plant and recently mentioned that it had small, green, spider-like bugs on it. She said that the last one she had got these as well and she was unable to get rid of them. My guess on these is that they are aphids. Is that probably correct? And if so, what's a good way to get rid of them on a Purple Passion?

Sounds like those pesky Aphids. I have used Insecticidal Soap on my house plants in the past. Seems to work for me. Don't soak the plant though, just give it a good spritz!!!!