Stupid - I am assuming you have a connection prior to LGA (as no direct long distance red eyes land in LGA). Shower - I'd suggest check the connecting airport lounge.

Regarding nap or sleep space - I hate to say it but if you plan to adjust to local time zone - you probably don't want to nap during daytime (especially if arriving from AsiaPac <assuming 14 hr flight>). You can take a 20 minute nap for functionality (even at airport) but encourage you to stay awake as long as possible until after sundown. Otherwise, it will be a more difficult adjustment.