I’m not sure about your base architecture, but one way would be to create a conditional field in the first table that is either empty or contains the desired status, depending on your schedule — something like
Then, in your remote table, use a lookup or rollup field to reference the conditional field you just recreated.
The mechanism is a little different from how you describe it: You seem to imply you want to push a value from your main table to your secondary table — which can’t be done in Airtable. (However, it is conceivably possible using a SaaS integration service, such as Zapier or Integromat.) Instead, what you want to do is pull a value from your main table into your secondary table. This is done using a lookup or rollup field, with the secret being to make sure the pulled field is empty until expiration of your schedule.