As long as your device is on the same platform, you can do this. Unfortunately, it's not possible to transfer purchases made on an Android device to an iOS device or vice versa.

If you're using an iOS device, please visit the game settings accessible through the main menu. There you'll find an option "Restore Previous Purchases...". Tap on it and enter the same credentials you used when making the purchase on your old device. As long as you're using the same Apple ID, your purchase will be restored for free.

For Android, simply visit your device's general settings and ensure you're signed in with the same Google Play Services account you used when making the purchase on your old device. The next time you start BitLife GO!, your previous purchase should be automatically restored.