iOS 16.4.1 stuck on preparing update on the setup screen

Check your WiFi Router / Modem for --> Recommended settings for Wi-Fi routers and access points ...

Connecting to Wifi is the first part and then getting connected to the Internet is the second

1. If you connect to Wifi, doesn't necessarily mean you are connected to the internet. If you are not connected to wifi could be a problem with iPhone settings or Router/Modem Settings
2. If you are not connected to the internet is the sole issue of Router / Modem Settings. Contact your ISP immediately

For your Internet connection.

• Over Cellular: contact your Carrier (Cellular Service Provider). The agency who issued the SIM to you
• Over WiFI: Contact your ISP (Internet Service Provider), the agency who provided the Internet Service on your Modem/Router to you

However --> Connect iPhone to the Internet - Apple Support

There are only 2 ways to update an iOS/iPadOS. Know how to...

• Update Over-the-Air (OTA) —> Update your iPhone or iPad - Apple Support (IN)
• Update using iTunes/Finder —> Update your iPhone, iPad, or iPod touch - Apple Support (IN)

Before any iOS / iPadOS update, please do the following

• You need to Identify your iPhone model - Apple Support
• It is also recommended that you either take a backup on iTunes or iCloud --> How to back up your iPhone, iPad, and iPod touch
• If you have a beta profile installed, remove the public beta by deleting the beta profile

Go to Settings > General, and tap Profiles & Device Management. Tap the iOS Beta Software Profile. Tap Remove Profile, then restart your device --> How to uninstall iOS beta software - Apple Support

VPN or proxy connections might prevent your device from contacting the update servers.

• Get help with over-the-air iOS updates - Apple Support
• Remove and download the iOS update again

If you still can't install the latest version of iOS, try downloading the update again:

1. Go to Settings > General > [Device name] Storage.
2. Find the iOS update in the list of apps.
3. Tap the iOS update, then tap Delete Update.
4. Go to Settings > General > Software Update and download the latest iOS update.

If you don’t see the update in the list of apps or if the issue happens again, you need to update your device with iTunes.

Try updating iOS after each step.

• Try to Forget the WiFi network and add it again
• Try another WiFi network 
• Toggle sets automatically off and then on
• Reset your network settings
• Force restart your device
• Remove any beta profiles, if applicable

• If everything fails: - Use iTunes or Finder (for macOS Catalina+) to update your device instead of over-the-air
• Get help with iOS updates and restore errors - Apple Support
• If you see an error when you update or restore your iPhone, iPad