My new iPhone 13 will not allow me to download the new update in stating that I’m “no longer connected to the internet.” Huh? It’s not letting me backup pictures on my Mac as a result. Any feedback? Thanks.

See the following Apple articles:

Update your iPhone, iPad, or iPod touch

Update your iPhone, iPad, or iPod touch - Apple Support (IN)

If your iPhone, iPad or iPod touch won’t update

If your iPhone, iPad, or iPod touch won't update - Apple Support

You can do the update via PC, see this Apple article: Update software on iPhone, iPad or iPod touch in iTunes on PC

Update software on iPhone, iPad or iPod touch in iTunes on PC – Apple Support (UK)

If you have a Mac see this Apple article: Update your iPhone, iPad, or iPod touch with the Finder

Update your iPhone, iPad, or iPod touch with the Finder - Apple Support

You need about 7 GB device storage and good Service & Wi-Fi.

If they are On, turn Off Hotspot & VPN.

It’s not letting me backup pictures on my Mac 

Check here -> How to back up your iPhone, iPad, and iPod touch

How to back up your iPhone, iPad, and iPod touch - Apple Support