I lost cellular service . Says to update software. But when I go to install it it says your not connected to the internet

It is asking for a carrier update not an iOS update.

Support article: If you see No Service or Searching on your iPhone or iPad - Apple Support

Go through the various suggestions in the support article above but if they don't help, and your network carrier can't help, then  contact Apple Support for further investigation. You should be able to reach them here

Contact - Official Apple Support

or phone numbers

Contact Apple for support and service - Apple Support

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