It is a fallacy that web sites might be unreachable for days just because the address changes. When done properly, nobody ever notices any downtime.

The proper way to do it is exactly what you did:
- make backup of existing server
- copy files to new server
- configure and launch new server
- change DNS to address of new server

It may take a while (usually 12-24 hours, typically not “days,” unless your domain configuration is different from most) for the change to propagate everywhere, and this is only because of address caching – whenever you request a web site’s address, the DNS servers don’t grab that info every time – they consult their caches, and return the address in the cache if it is there. If the address is not in the cache, web users will see the new site the instant you flip the switch.

The cache ages, and the address goes away after a time. In fact, you control that time – it is known as the TTL (tme to live) in the SOA (start of authority) record of your domain name configuration. Your provider probably gives you access to those settings in an advance configuration screen.

Cheers.