In a kitten this young, you're likely dealing with some type of virus. That could explain the vomiting and lethargy. The lethargy is usually due to a fever. These will usually run their course over 3-5 days.
To help with the vomiting, there are a couple over the counter meds to settle the stomach:
1. Famotidine (Pepcid)- you can give 1/2 of a 10 mg tablet every 12-24 hours
2. Meclizine- you can give 12.5 mg every 24 hours for nausea.
Other than that, you can try to entice him to eat with canned food. Warming it up in the microwave can also help to get his appetite back.
If the vomiting worsens, then I would start to worry about a foreign body obstruction. In that case, you really need x-rays and surgery to remove it. It's less likely, but something to keep in mind.
I hope this helps, let me know if you have any other questions.