How to get around changing information payment in obsolete iPhone

You would have to use newer equipment, or a web browser. This document explains how to add a payment method to your Apple ID on a Mac, Apple mobile devices, PC, or online --> Add a payment method to your Apple ID - Apple Support

However, even then a 3G might now work since apparently Apple cut off store access to a whole line of older equipment and systems. If you can't open the App Store or iTunes Store, buy content, or edit payment information - Apple Support - On May 25, 2018, Apple introduced security changes that prevent Windows XP or Vista PCs and Apple TV (1st generation) from using the iTunes Store. OS X El Capitan at least will be required on Macs. Any hardware considered to be obsolete [ Obtaining service for your Apple product after an expired warranty - Apple Support ] may not work.

but I wonder if there's any workaround.

There's no workaround.

iPhone 3G (iOS 4.2.1) is nearly > 15 years, which also affects the battery life. The hardware remains old, so does the battery.

It is no longer possible to have the battery replaced.

If you have an obsolete iPhone that cannot be updated any more, many apps will not be able to download.

Obsolete iPhone no longer meets the requirements for many apps.

See -> iPhone products obsolete worldwide

• iPhone
• iPhone 3G (China mainland) 8 GB
• iPhone 3G 8 GB, 16 GB
• iPhone 3GS (China mainland) 16 GB, 32 GB
• iPhone 3GS (8 GB)
• iPhone 3GS 16 GB, 32 GB
• iPhone 4 CDMA
• iPhone 4 CDMA (8 GB)
• iPhone 4 16 GB, 32 GB
• iPhone 4 GSM (8 GB), Black
• iPhone 4s
• iPhone 4s (8 GB)
• iPhone 5c

Most apps require iOS 13.4 or later.                                

Considering a newer type of iPhone?

👉 Read this before you buy a 2nd hand iPhone.

Most apps require iOS 13.4 or later.                                

Considering a newer type of iPhone?

Read this before you buy a newer type iPhone.

You can try using "none". I don't have a payment method set on my account. You mostly need it for getting certain apps, but since you may possibly find there are absolutely no apps still available for your 3G then it probably won't matter. :-)