Trying to recover a file on AirPort

Since your Mac is running macOS Sonoma, and not Sierra per your tagline, when the TC is not connected to your Mac, TM will continue to run hourly backups to your Mac as "snapshots." These would also appear in the TM timeline. Note; however, that these snapshots typically deleted daily to prevent you Mac's internal drive from "filling up."

Let's see if those snapshots still exist. To do so, we'll use the Disk Utility (DU).

1. Run the DU. It's located at /Applications/Utilities.
2. From the macOS menu bar, with the DU running, select View > Show APFS Snapshots
3. A new window will appear at the bottom of the DU.
4. If not already, change the DU View to: Show All Devices
5. On the left-side window, select "Data"
6. In the following image, I placed an arrow pointing to the snapshots.

Do any appear for you? If so, then you can see the dates that they were taken. You would see that they only go back a few days, as again, they are purged often. If that "important" file is from an earlier date, then unfortunately, this won't help you.

Instead, you may need to consider soliciting the help of a dedicated recovery service in your area to try to recover your file(s).

You may also want to consider retiring your 3rd generation TC. At least, not to continue to use it for TM. This device was discontinued in 2011. Most consumer-grade networking hardware typically have a useful life of 5-7 years. The fact that you had this one operational this long is a rarity.

One option, with the Time Capsule (TC) connected, is to use the Finder app to try to locate that important file.

To do so:

1. Open a new Finder window.
2. From the left window, select your TC. It should appear under the Shared section.
3. Click on Connect as... and enter the TC's disk password, and then, click on Connect.
4. Double-click on the disk partition to open it. Note that the partition should also mount onto the macOS Desktop.
5. Any available backup .sparsebundle should now appear. Select it.
6. One of three things should happen: 1) A "folder," titled: Backups.backupdb, should appear, 2) The "Time Machine Backups" drive will mount on the macOS Desktop, or 3) "Time Machine Backups" will appear under the Devices section in the Finder.
7. Double-click on Backups.backupdb to open it. Look for the folder with your computer's name on it.
8. Open that folder and you should find a series of dated subfolders.
9. Open any of those folders which you believe where the last time TM saved that important file.
10. You should now see your Mac's basic folder structure where you can pan down to where you thing it may exist.
11. Once you find it (hopefully), you can then copy it to the macOS Desktop.

Just to confirm ... is your iMac, that you are trying to back up to this TC, still running OS X Sierra (10.12)? Also, which exact model is your TC? There were a few of them that had 2TB HDDs.

Can you see the TC in the Finder? What have you tried, so far yourself, to resolve this issue so that we don't offer potential solutions that may not have already worked for you?