does adobe illustrator is compatible with my MacBook Air ModelA1369 it has El Capitan SOftwere

Depends what version of Adobe Illustrator you are using.

If it is a new version then it probably does not work with El Capitan.

You may need to upgrade your OS.

The App Store Updates page only informs you of updates available for the OS you are running.

Installing a newer OS is called an upgrade.

Which OS you can upgrade to will be governed by which model and year your mac is.

Click on the  top left of your screen click on About This Mac to find out the year of your mac.

If you find your mac can upgrade to the newer OS's like macOS Mojave 10.14, macOS Catalina 10.15, 

macOs Big Sur 11.4 I would recommend you upgrade to macOS High Sierra 10.13 first.

For older OS’s,

Open Safari and click on this link, 

How to get old versions of macOS – Apple Support

Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.

Then go to the Download OS section and click on the OS you want to install.

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.

Connect your mac to your router via cable rather than rely on WiFi.

Do not use your mac while it is downloading

and do not use any other devices that share the same internet connection.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS ******.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

If you want to or are able to install Big Sur, open Safari and click on the link below.

How to upgrade to macOS Big Sur - Apple Support

If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.

Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,

some updates may be free, some may be paid. Use this app to discover which apps you have

are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see

if they are offering 64bit updates.

There are no Illustrator upgrades if you are still using one of the ancient CS5/6 suites. Currently supported versions of Illustrator for macOS require a subscription and only Mojave thru Big Sur are supported with these subscription versions.