Show last listing update time next to connection status in header

Add last_updated timestamp to /api/status endpoint by querying
MAX(last_seen) across both listing tables. Display it in the
HealthIndicator as relative time (e.g. "2h ago") with full
date/time in the tooltip on hover.

repositories/listing_repository.py

@@ -359,6 +359,22 @@ class ListingRepository:
 
         return model_listing
 
+    def get_last_updated(self) -> datetime | None:
+        """Get the most recent last_seen timestamp across all listings.
+
+        Checks both RentListing and BuyListing tables and returns the latest.
+        """
+        with Session(self.engine) as session:
+            rent_max = session.execute(
+                sa_select(func.max(RentListing.last_seen))
+            ).scalar()
+            buy_max = session.execute(
+                sa_select(func.max(BuyListing.last_seen))
+            ).scalar()
+
+        candidates = [t for t in (rent_max, buy_max) if t is not None]
+        return max(candidates) if candidates else None
+
     def get_listing_ids(
         self,
         listing_type: ListingType = ListingType.RENT,