A 4 month old calf being weaned off a cow is very unlikely to drink milk replacer from a bottle or a bucket. He needs to be penned separately, in a small enough pen that he can't get away from you too far and that you can monitor him. Feed him a good calf grain and good hay and water. He should be kept separate from the rest of your herd for at least a week or 2. Make sure he is eating good, watching for runny nose or any respiratory problems. Then vaccinate with whatever your normal protocol is
Yes he is going to bawl. He is going to be unhappy being taken away from his mother. If you have a real friendly one the same age that is weaned you could put it with him for company and hopefully neither will get sick.
We do not give antibiotics to anything we bring in, but we do vaccinate for blackleg and use a killed virus vaccine to help boost their immune systems. Anything new is kept close for a week to 10 days to watch for problems before they intermingle with anything else.