Flying coast to coast Non-stop from LAX to DCA in First Class, does this qualify for entry into the Flagship Lounge? If not, is a one to two hour pass available?

As Luis says

Only Admirals, if going to BOS or MIA you can use Flahship

No only real transcon flights are eligible for Flagship Lounge which means LAX to JFK or BOS or MIA

Usually cross country flights are eligible for entry as a convenience to the Admirals Club. DCA does not have listed any Flagship lounge however. Check at the airport to be certain.