I'm assuming that you have created DNS zones for your website. You don't need to have your domain DNS pointing to the new server, but you need to create a DNS zone to open that website with only IP on your machine. How to do that is out of the scope of your question.
You could access your website only with an IP, It doesn't matter if IP isn't dedicated. What you have to do is to specify inside your HOSTS file in your machine that IP (VPS or some other not dedicated IP). You specify where your websites files are and that there is an IP for that domain where to go to get HTTP responses like so:
Open your HOSTS file on a unix system could you probably find it and modify it from terminal by:
# vi /etc/hosts
There add your new host e.g. for an IP 188.8.131.52 where yours website's files are. Add this line at the end of file:
Now you have to just open your browser and try to get this url:
You will get you development website files on your browser in your machine.