Can budget for member account be managed from Payer account.

You absolutely can do it. We do have documentation on Budgets however it works best when you just try it out in practice. Creating budgets is free (you only pay for your budget if you use Budget Actions feature), and you can easily delete the budgets you don't need later.

I suspect that you were creating a budget through the template path. I suggest to customize it (advanced). When creating cost budget, at the bottom of the settings page you'll have the Budget scope. Select "Filter specific AWS cost dimensions" there. And now you can apply filters in a same way as you would, if you were just checking your cost in Cost Explorer, including cost by linked accounts:

You can either create separate budgets for individual linked accounts, or for groups of accounts. When setting up notifications, you can then send those notifications (when the account is about to reach the budget) to the relevant team email or distro list. Here's more info in the docs regarding customizing budgets: https://docs.aws.amazon.com/cost-management/latest/userguide/custom-budgets.html