JAVA - Validating User inputs on service methods in a cleaner way

I'm not sure what you're getting at about it seeming odd. I suppose the various if/throws are repetitive and could be reduced by writing a function for them.

private void requireNotNull(boolean requirementMet, String errorMessage) {
    if (requirementMet == null) throw CustomHandledException(errorMessage);
}

// or, depending on whether the messages require any calculations:

private void requireNotNull(boolean requirementMet, Supplier<String> lazyErrorMessage) {
    if (requirementMet == null) throw CustomHandledException(lazyErrorMessage.get());
}

This is getting really nitpicky, but I think the logic flow would read slightly easier if you handle the non-null case of getUserResponse first so you don't have a big chunk of code in between retrieving something and working with that retrieved thing.

public void exampleMethod(final UserDto user, String sthElse) {
    requireNotNull(user.getId(), "example");
    requireNotNull(sthElse, "example");

    GetUserResponseDto getUserResponse = client.getUserById(user.getId(), country);
    if (getUserResponse != null) { 
        Locale currentLocale = getUserResponse.getUser().getProfile().getLocale();
        if (!user.getLocale().equals(currentLocale)) {
            client.updateUser(user);
        }
    } else {
        requireNotNull(user.getGivenName(), "example");
        requireNotNull(user.getEmail(), "example");
        CreateUserRequestDto createUserBody = UserMapper.toCreateBody(user);
        client.createUser(createUserBody);
    }
}

Aside from that, I wouldn't start variable names with verbs.

Separate into any function

private void validateInputs(final UserDto user, String sthElse) {
    if (user.getId() == null || sthElse == null) {
        throw new CustomHandledException("example");
    }
    
    if (user.getGivenName() == null || user.getEmail() == null) {
        throw new CustomHandledException("example");
    }
}

private void handleUserCreation(final UserDto user) {
    CreateUserRequestDto createUserBody = UserMapper.toCreateBody(user);
    client.createUser(createUserBody);
}

private void handleUserUpdate(final UserDto user, GetUserResponseDto getUserResponse) {
    if (!user.getLocale().equals(getUserResponse.getUser().getProfile().getLocale())) {
        client.updateUser(user);
    }
}

And then combine it in:

public void exampleMethod(final UserDto user, String sthElse) {
    validateInputs(user, sthElse);

    GetUserResponseDto getUserResponse = client.getUserById(user.getId(), country);

    if (getUserResponse == null) {
        handleUserCreation(user);
    } else {
        handleUserUpdate(user, getUserResponse);
    }
}