How can I update my iPad mini without buying cable connections?

The iPad Mini 3 cannot be updated beyond iOS 12.5.4.

If yours is not running 12.5.4, see the following instructions: Update your iPhone, iPad, or iPod touch - Apple Support

If you need to update manually using a computer, you will need a cable. The cable that you use for charging is what is usually used.

Hello. The maximum version of iOS your iPad mini 3 can run is iOS 12.5.4. If you are trying to update to a version of iOS newer than this, it is unfortunately not possible on this device. If you would like to run iPadOS, such as iPad OS 15, you will need to upgrade to a newer iPad.

That being said, if you are trying to update to iOS 12.5.4, you will need to find a reliable wireless access point or connect your iPad to a Mac or Windows computer.

If NO access to a Mac or Windows computer, you will have to seek help from a good friend or family member to help you out

You need the cables you already have to charge the iPad.

OR

Search online to locate and call contact your nearest, “local” Apple Store location to try/attempt to make an appointment to get your Apple iPad upgraded/updated to its latest iPadOS 14.6.

Apple Store employees will be glad to help/assist you.

Best of Luck to You!