A.W.
Here's the good news: a 4 week-old is 100 times easier than a six month-old! It's much simpler when they aren't mobile. My baby has been on several flights since he was born (mostly short trips to see Grandma in Northern CA), and he's always done great.
1. Formulate a game plan to get through the Security check point! Everything has to go through the x-ray machine, including the folded stroller and car seat. You can bring formula through, but it has to be pre-prepared. If you're bottle-feeding, invest in the ready-to-feed bottles-- it's worth the expense.
2. They always call "families with small children" to board the plane first, but don't do it! The longer you spend OFF the plane, the better.
3. Ask the flight attendant for extra pillows to prop up your elbows and/or the baby while breastfeeding.
4. Very few plane restrooms come equipped with infant changing tables, unfortunately. Chat with a flight attendant before you sit down to determine a game plan for baby changing. I once had a very sweet flight attendant let me use the floor in their seating area to change my baby. Better there than in the aisle!
5. Bring infant Tylenol. The cabin pressure didn't bother my baby until he was 5 months old, but it's better to be prepared! If possible, try to nurse your baby during taxi and take off-- it helps relieve the ear pressure.
6. On one flight, I brought tons of ear plugs and gave them out to the people in the seats around me as a peace offering. Inevitably, your baby will cry. Do your best, and keep a sense of humor, apologize sincerely, and you'll be fine.
Good luck!