Uh, I suppose if I had to save someone I would try and find out why they don't want to live, then go from there. Being able to truly listen with a sympathetic ear is important. Being able to let go of biases that prevent you from really being present with the person and their story, their struggles. That alone is hard work and many people just don't have the capacity to do that. Hell, I don't know if I can do those things myself...but I can try lol.
Sometimes a person's desire to die is rather complex and there's not much that one average person can really do to "save" them. I guess I would try to help them find resources if they're in a really bad situation - good therapy, shelters, etc. Maybe they need to hear from someone who has walked a similar path, shared similar struggles, and survived. Perhaps offering a different perspective can help.
I feel like it would take a lot of time to truly save someone, the above is very idealized tbh. Even then, after time and effort, there's no guarantee that the person will find a desire to live. There's no guarantee that they'll find hope. And, really, in the end the choice is up to them. I wouldn't feel comfortable talking to someone who is suicidal with a goal in mind to save them, because that's not up to me.
I hope this made sense. Spent a lot of time trying to get my brain going and writing something is coherent.
Just realized I didn't answer your other questions. First, you asked if someone must have purpose to live. I don't know about purpose, but I think it's important to have desires and goals. Without those things, life can be rather empty. People talk about purpose a lot and I think it provides comfort and meaning, but I don't think it's a necessity. You also asked how one would live without hope... People definitely survive without hope. Generally what keeps them going is something like fear or a sense of duty/obligation. It's not a desirable way to live. :(