I just went through this! We just took a trip to Florida with our almost one-year old. I have to say that if it is at all affordable to buy an extra seat for the baby, do it. Part of it is I have a big baby, so if your baby is smaller it may not be as difficult. We kept him on our lap, but he is so long that he kept kicking the lady next to me, not fun. Plus we weren't able to put down either of our tray tables because he covered my full lap and half of my husband's when he was laying down.
Everyone is going to tell you to try to give the baby a bottle on take-off and landing. This is good advice if it works for you. For us, we could never time it quite right and the baby was generally demanding a bottle by the time we got on the plane, which is well before it actually takes off. Plus, the descent usually takes about 20 to 30 minutes where it is affecting your ears, and my child can down a whole bottle in under 5 minutes, so it didn't work for me. Luckily, his ears seemed to make it through okay despite our failed attempts and the fact he had a cold.
I brought with a couple of new toys that he had never seen before so that he would be interested, I packed a TON of different snacks for him, and a couple of his favorite toys. Plus, it is so important to have your diaper bag organized precisely so that it is really easy for you to find exactly what you need in a tight space.
I also highly recommend the Sit n Stroll which is a car seat that converts into a stroller. We found it invaluable on our trip because we didn't have to lug both a car seat and a stroller with us everywhere. The Sit n Stroll is the only thing like this so even though it's not perfect, it is still a great travel tool.
Lastly, just try to relax. Everything won't be perfect and it is possible that your baby will have a meltdown on the plane (mine did). But most of the people around you will be really understanding and won't be nearly as upset about it as you are.
Good luck and enjoy your trip!!
L.