How do I invite friends to my business page with my personal account?
You can't. Your Facebook Page has nothing to do with your private profile. In most cases friends of your private profile are not going to even discover, that this particular Facebook Page is yours (unless you of course tell them this). You may try to like your own Facebook Page, when you're login as your regular, private profile and then invite your friends to do the same. But you can always do this only if you're logged in as private profile.
Facebook Pages are meant to represent business, events, campaigns etc. That's why they can't have friends, can't like, share or comment private profiles (only other Pages) and are much more limited.
Can I update the business page when I am logged in under my personal?
Of course! If you're particular Facebook Page's admin you can manage every aspect of that page, no matter, if you're logged in as that page or as a regular, private Facebook profile.
If you use Facebook gear icon and click
Use Facebook as [profile name] you'll be redirected to your private profile. Then you should press Back in your browser or on your mobile device, to go back to your Facebook Page or opening and you have there access to all the tools, like you would have once logged in as that page.