Early Arrival

Hi, My wife and I leave Sydney at 9.30pm and arrive into Haneda Airport at 5.05 am the following morning. Our check in is at 2.00 pm in Shinjuku. I am toying with the idea of going to the hotel and dropping the bags. This should take approximately 2 hours and then go to Hakone on the 2 day pass, only using one day, from Shinjuku and arrive back at the hotel at about 7.00pm to 8.00pm. Do you think this is too much for the first day.

Well, it depends on how good you sleep on planes. I don't at all, so it would be a total no go. Frankly, I would just stay in Tokyo and do some Tokyo sightseeing. That way, if you get tired you can be back at your hotel at 14:00 and rest. And if you don't, you just get your planned Tokyo sightseeing done.

Even though there's no jetlag issues due to a similar timezone, I can't imagine having a big day after the early morning start and being on the plane overnight. I know when I fly out of an evening from Tokyo and arrive in Australia about 6:30am, I can barely function all day. Your idea of dropping off the bags is perfect, but I think you should take it easy in Tokyo that day. Shrines, temples and parks/gardens open early, so it would make sense to go for a stroll to one or more. For example, Meiji Shrine (short distance on the Yamanote line from Shinjuku), walk to Shinjuku Gyoen from your hotel, catch a train to Asakusa to see Sensoji, etc.

I have somewhat different opinion.

This should take approximately 2 hours and then go to Hakone on the 2 day pass, only using one day, from Shinjuku and arrive back at the hotel at about 7.00pm to 8.00pm. Do you think this is too much for the first day.

You will arrive Haneda at 5:05 which means you will not hit the morning rush hour. If you take monorail and then JR, it will be less than 50 min to Shinjuku. You may need to walk to your hotel, but it will take around 1 hour from the airport to the hotel. Then if you catch "Romance car" limited express, which will take 90 min to Hakoneyumoto. Ninety minutes ride in comfortable train is not so hard even if you fly 10 hours. Rather, 90 min ride on the Romance car will be one of the easiest trip: you don't need to walk, you can even nap on the seat.

I think you want to see Mt Fuji from somewhere in Hakone. If it is winter, your chance will be more than half. But of course it depends on the weather. You don't need to pre-book the Hakone free pass, so you might better check the webcam of Hakone to foresee whether you are likely to be able to see the mountain. If you are in good condition after the long flight and the weather in Hakone area is fine, and you want to do the trip to Hakone, then just go. I recommend you to take the limited express for your comfort. Of course you should be prepared for the plan B as has been already suggested.

I think Hakone is doable, but it will be a very long day. Honestly, it is a very long day even if you have had a full night's sleep!

Personally, I'd just go check your bags, have a nice breakfast/coffee and then just take in some morning sights in Tokyo before heading back to the hotel at 2pm for a short nap.

Go and drop you bags off at the hotel and enjoy some light sightseeing in Tokyo.

At least if you feel a little down by the middle of the day, you'll have a bed available by 2pm.

No good risking felling tired in the middle of the day at Hakone and the having a poor experience.

A long day and do-able, especially if you sleep on the plane and again on the train (if tired).
Might be more than two hours from landing to clear immigration/customs, head to the city and drop bags and then get to the Odakyu counter to buy the passes. But even so, that is ample time to get to and around Hakone for the day.