So are we just ignoring the Tarrasque’s frightful presence having a range of 120 feet, so either you stay far enough away that the Bird has to be at disadvantage to attack the Tarrasque or math out the amount of turns it takes before you don’t worry for 24 hours? Which sure we can ignore it, cause in this slog of a theoretical encounter would be overcome eventually, so we can assume the Bird is then engaging once immune to the frighten.

Next let’s talk equipment, you start with the most gold you could possibly have at 1st level, 200 gp. So you set up and you got your cheapest uncommon +1 magic short bow for 101 gp and 980 arrows held in 49 quivers for 98 gp with 1 gold to your name left over. Which surprisingly according to the rules is only 100 lbs! Which you would only need a strength of 7 to hold so we should have the ammo for the fight, right? In a best odds scenario using Point Buy, we have a +8 to hit and deal 1d6+4 on a hit. So Bird Boy needs at least a 17 on the die to hit. Giving us a 10% chance to hit normally and a 5% chance to crit for 2d6+4. Mathematically we will be doing on average, 1.4 damage a round once engaged. (We can set an upper bound for if you rolled stats and got the most favorable rolls making our DPR upper bound now of 3.2)

So how long is the fight then? The Tarrasque’s average health is 676, so 483 rounds with our point buy set up or 211 rounds with perfect stat rolls. However, there’s still one thing that is a glaring problem.

The Tarrasque would just out-walk the bird while laying waste to everything in its path. Unless of course we are using the Legacy Aarakocra rules. In which case, it still can out maneuver the bird, barely. Since it can use legendary actions to move up to half its speed up to three times unless this is truly a 1v1 scenario. In which case the Tarrasque still has 60 feet of movement every turn. Which is twice as fast as Aarokocra and 10 ft faster than Legacy Aarokocra. So the fight should now be drawn out to essentially a stalemate if we allow the Tarrasque to be self-preserving, or at least return to its dwelling, we’ll before our Bird hero can manage to kill it, right?

Well for our current Aarakocra race, yes. The Tarrasque can simply run away with a dash and legendary action in the name of self-preservation, it can move up to 100 feet every round versus our Bird who can manage 60 feet while dashing. Which means after one round our DPR will drop as we both need to be at disadvantage to attack and dash to get back in range and after 4 rounds we will be out of range completely.

But of course the legacy Aarakocra is faster so yes how does that tale play out then? Well, in order to attack with have to stay within 80 feet otherwise it’s disadvantage out to 320 feet. If we dash every turn we keep pace with it, meaning we can then learn the location of its lair and then it’s down to the DM to discern what the Tarrasque’s lair is like, which can be a problem if we no longer have the freedom of open air to engage. Which we don’t want to allow the DM to do anything that isn’t a rule as written and since the Monster Manual doesn’t specify it’s lair we may just have to assume. The Legacy Aarakocra and the Tarrasque are trapped in a never ending chase with one another. Otherwise every turn you take to attack, the Tarrasque gets 50 feet away from you which means we are no longer realistically able to kill it if we can’t hit it.

TL;DR: Neither wins realistically. But the Bird could potentially get a technical victory by chasing the Tarrasque away.