I do not have Apple Store account, can I update my Safari without it?

To upgrade Safari, you need to upgrade the OS X version to the latest version from App Store.

Sign in to the App Store, download and install the OS X version if your Mac meets the system requirements.

The newest version of OS X (macosSierra) will be released soon.

Just use your Apple ID to download OS X (macOS).

For more info:

https://www.apple.com/support/mac/app-store/getstarted/

If you are posting here, you have an Apple ID, which means you can sign into the App Store, and complete what dominic posted.

What OS version and Safari version are you using now?

Check to make sure your applications are compatible. PowerPC applications are no longer supported after 10.6.

Application Compatibility

Applications Compatibility (2)

El Capitan 10.11 Compatibility information

Also check to make sure there is a compatible driver for your printer.

Do a backup before installing, preferable 2 backups on 2 different drives.