Obviously, this is going to involve some luck, but there are a few things you can do to tilt things in your favor. I recommend playing against one Automa and three AIs; I set mine all at medium difficulty, but I'm not sure how much difference that makes. Having the maximum opponents increases the chance that you'll be given free resources, which can free up your turns to focus on point-getting activities. And the Automa periodically fires a "trigger all pink powers" event, which can really help if you get an early vulture or something. Egg laying is an important piece of any high-score round, but it's not going to be enough to hit 120. So, you should get some birds in your middle row that let you tuck cards. It seems like you should be able to get more points with the "non-competing" goals, because you could theoretically get 20, but I found I did better with the "direct competition goals".
My winning board (140 points) had a Ruby-Throated Hummingbird, Chihuahuan Raven, House Finch, Canada Goose, and Northern Mockingbird in the middle row. That let me get food every time I laid eggs, and tuck five cards. In my bottom row I just tried to play high value cards (Osprey, Pied-Billed Grebe [for cards, not points], Bald Eagle, Atlantic Puffin, White-Faced Ibis). I had no birds in my top row.
For reference, my points breakdown was: 46 from birds, 14 from bonus cards, 14 from round goals, 27 from eggs, 0 from cached food, 39 from tucked cards.
I did not test it for this achievement, but other achievements can be earned in games with multiple human players, so it's possible you could play multiple "hands" and try for the high score with your "primary player" while the others "feed" it by playing cards that benefit all players, and intentionally triggering pink powers. I would probably still have the Automa as one of the players because of the "trigger all pink powers", though if you played all players, you could intentionally not compete for the end round goals...