Managing two-factor authentication abroad without a local phone number

First, normally you should only rarely have to receive a verification code.

If I understand you correctly this will be easy as long as you have a second item of Apple equipment signed into your account whereby it can receive a verification code. The below is based on this support article Get a verification code and sign in with two-factor authentication - Apple Support

A code is sent automatically first to your trusted devices (some other Apple computer or Apple mobile device signed in with your account). If you have a trusted device running iOS11 (or iPadOS) and later, macOS 10.13 (High Sierra) and later, or watchOS 4 and later, the verification code is displayed automatically on your trusted devices.

If you don’t have a trusted device handy, you can choose to have a verification code sent to your trusted phone number as a text message or phone call.

1. Select “Didn’t Get a Code?” or “Can’t get to your devices?” on the verification code screen.

2. Choose a telephone number for where to receive a code.

3. You'll get a text message or phone call from Apple with your verification code. This text message might include an additional domain validation line. This line includes the @ symbol, the website name, and your code (for example: @icloud.com #123456 %apple.com).

4. Enter the code on your other device to complete sign in.

If you only have one item of equipment with you the process becomes more complicated and can be helped by having a recovery contact (if you know somebody you trust in your new country you could even set this up right now).

Set up an account recovery contact --> Set up a recovery contact for your Apple Account - Apple Support

- "An account recovery contact is someone who can verify your identity and help you regain access to your account and all of your data if you ever get locked out."

- You can select up to five recovery contacts for your account.

- Your contacts will need an iOS device with iOS 15 or iPadOS 15 or later, or Mac with macOS Monterey or later

- If you forget your Apple ID password or get locked out of your account, just follow the steps on your device to share the onscreen instructions with your recovery contact in-person or on the phone. Your recovery contact then shares a six-digit code that you enter on your device. After your identity is confirmed, you can reset your Apple ID password.

- "All of your devices must be running iOS 15, iPadOS 15, watchOS 8, or macOS Monterey or later. You might be prompted to update any devices not running the latest OS, or remove any devices that you aren't using."

The best way to manage two-factor authentication (2FA) before getting a local phone number abroad is to transition away from SMS-based 2FA to authenticator apps, security keys, or temporary virtual numbers. SMS-based 2FA is highly vulnerable to losing access when you change SIM cards or travel outside your home carrier's network coverage.