When does season 2 drop?

Sometime this year. Maybe.

By the time Season 2 actually appears, I’ll be the Old Man

I think they said sometime in Fall 2024.

Is there going to be a second season?

Seems to be two issues here ... from Wikipedia:

Shooting the series was two-thirds complete when production shut down in mid-March 2020 due to the COVID-19 pandemic.[20] Production resumed in late 2020 with three episodes remaining to film,[20] but was soon halted when Bridges was diagnosed with lymphoma in October 2020 and departed for cancer treatment.[21] The show stayed in production for a short period after his diagnosis to shoot scenes from the remaining episodes that do not feature his character.[20] In September 2021, he announced his cancer was in remission. Production ultimately resumed in February 2022.[22] On June 27, 2022, the series was renewed for a second season.[4] In May 2023, it was reported that filming of the second season was suspended due to the 2023 Writers Guild of America strike.[23]

Hard to figure out what is really going on and when we might see some new episodes. This was an interesting show, I'd like to see it concluded.

https://deadline.com/2024/07/the-old-man-season-2-premiere-date-fx-1236005616/