3 month old was pretty easy - in my experience seems like the more tricky time is around 1-2 when they are very mobile but can’t be distracted for long, nor bargained with.
I did Oz-NZ several times by myself with an infant, and then infant + toddler and the anticipation is definitely the worst - but take the time to prepare and it should be better than you expect. They days of reading a mag or watching more than 5 minutes of the movie are long gone though!
I planned flights sort of around nap time - I didn’t bring a car seat or anything like that, preferring to have less bulk to haul around. I used a soft Mai tai sling which I could carry her through the airport in, and then let her sleep in. I nursed during takeoff to help with ear pressure (takeoff and landings worst for screaming episodes) and this plus timing would often send her to sleep.
I sometimes got the option of a clip on bassinette in front, but in all cases she had to be sitting on my lap for takeoffs and landings with a mini seatbelt (i didn’t have to pay for another seat this way).
On checkin I always asked for a seat away from other people (unlikely during the holidays) but this often got me a spare seat next to me where I could lay her down to sleep. If not, when she was sleepy I’d pop her into the sling, go and stand in a clear section of floor and jiggle her until she went to sleep. Then I could sit down. Aisle seat is best, can get up and away if need be.
I would bring extra nappies (got a flight delay of several hours one time), change of clothes, a pashmina or something to block out light and distraction, a chewing toy (again for ears), a couple of other quiet toys, a sleep comforter if the baby uses it, plus baby Panadol (painkiller) if you think the baby’s ears are playing up).
I struggled to eat the plane food with her close by (if you have a separate seat and car seat this would be easier) - so brought snack food I could eat so I didn’t get crazy hungry.
The flight crews were uniformly great, a single woman with kids gets a lot of breaks at least on Air NZ and Qantas flights. I got escorted through customs, helped onto planes, had pushchairs waiting for me at the gate, crew offering to carry the baby around for a bit etc. if you’re travelling alone make sure they know and someone should keep a little eye on you.
Above all, this is not going to be a relaxing flight. The baby will cry, but likely not as long as you think (or it feels!) Most people won’t care, they have their headphones in. What will bother people is you clearly doing nothing to remedy it. If you are managing the situation, doing what you can to get through the flight, it will all be fine.