Bit rubbish that your host only allows one database.
Anyway. . .

1. Copy your old database and download it from your site to your computer. (Keep it there.)
2. Create new database on your server for new install.
3. Do fresh install.
4. Delete all tables from new database and import the old database to new install.

Not sure if it is the best procedure but this is what I would have done in your case and may have to do myself if 5.2.0 is messy.

I don't have a lot of knowledge about database , so that's why I can't do that , I don't want to lose data . Still waiting for professional assistance from vbulletin staff team , but there are no answer, ,,, I think they answering when is about money... hope they will answer me and fix it today.

Well this is not really something vBulletin should fix for you. They can advice you but they are in no way obliged to help you out with this. Your best bet is to hire someone to fix this problem for you.

Yes , but I'm still customer, so they should to provide some help , as I mentioned that is from new version issue. If they can't fix for me , is fine I'll fix , but IDK how .

You don't need any knowledge. I don't have knowledge on database issues either.
It is all done from your server. cpanel takes you to php MyAdmin and all database creation and deleting, importing, copying etc is done from there. Even if you learn how to copy your database, you are already doing one of the most important tasks that you will need forever.

COPY your database and download it now!

Ok can you explain please how to do step by step ? BTW , if installation will gonna be successful and I will want to put old database, how can I do that ?

Do you have cpanel on your server?

Which ? I have phpadmin panel , and ovh cpanel

Go to phpMyAdmin panel.

ATM I'm at work, so I can't do anything now , but can you explain please step by step , hope I can fix that , as I see support from vbulletin is really badly.