For S-Corp business filing 1120-S, My start date is 03/16/2024, but I cannot enter start-up cost deductions because it says this is not my first year doing business.

the issue is that you were a c-corp from 2/13/2024 until your S-corp election was effective on 3/14/2024.

thus it would seem all start-up costs would be C-corp costs. 

item E on the 2553 should have been 2/13/2024

technically you are supposed to file a C-Corp return for the period 2/13-3/13

from 1120 instructions Who Must File
Unless exempt under section 501, all domestic corporations
(including corporations in bankruptcy) must file an income tax

return whether or not they have taxable income.

I don't have a solution for your issue. 

perhaps others will have different opinions on how to resolve this issue