use for example a passwort protect plugin like this:
https://wordpress.org/plugins/knock-first-password-protection/

You could even create a new LocalWP site, then use one of the plugins i mentioned above to copy your online site over to a local site.

And then build your new site online, replacing the old site.

This way you would still have your old site as reference locally!

I think if you're happy with siteground it is easier to stay there.

Otherwise you'd have to either configure the domain to point to netcup using DNS (domain name server) configuration, which is quite technical.

BUT, i think you can even get hosting at netcup, then ask them to do the domain transfer for you:
https://www.netcup.com/en/domain/domain-transfer

And then transfer the site mmh.

However, if the price change is substantial, it might be worth the hassle to move over. But doing that can be quite technical and I think many people would be a little deterred by it. However, some webhosts offer "moving services" im no too sure to what extent netcup offers this.

In your position I would do it like this:
Install LocalWP, then create the website with LocalWP on your own computer.
When you are done, to move it online, use this plugin
https://wordpress.org/plugins/all-in-one-wp-migration/

or this one
https://wordpress.org/plugins/backup-backup/

read more here:
https://laytheme.com/documentation/moving-website

if you have the ChatGPT desktop app, or Claude Desktop App with a subscription, AI can even help you do that and move the site for you, using one of these plugins or some other plugin that it decides could be optimal