To Shiodome from Haneda airport: I would take the Keikyu train to Shinagawa. There you can possibly buy your Narita Express tickets for the next morning, if the ticket office is open (there are also shinkansen/limited express ticketing machines that can be set to operate in English if you touch the "English" thingie). Then take the Yamanote line to Shimbashi station and walk to the Conrad from there.
The ticket machines selling shinkansen/limited express tickets are usually in an area of the ticket machines colored a light purple-ish, I believe. But that's marked in English too.
Note that you will need Yen cash for the N'Ex tickets from the machines (around Y6000-6500 for two). The JR ticket offices take credit cards but not the machines.
I was in the ANA F lounge last time I was at NRT, you can make a meal there but it's not as much better than the business lounge as I had expected.