I am already using my domain

Hi,

I have a website running on a host. I want to build a new website with WordPress on this host using my current domain name. I have looked and looked…however I can’t find if it is possible to first build a new website before it goes online meanwhile still keeping my old website up and running untill my new site is all ready to go.
Is there a possibility for both my old site to run and meanwhile build a new one with WordPress? Without my site being ‘under construction’?

Is there a possibility to try out some Themes and ideas on my computer before the go online?

Thanks!!