1) Could be a problem in case the train is full.
2) The return ticket from Narita Airport to Beppu is 50,020yen for Narita Express, Shinkansen Nozomi and Sonic rides, with the validity of 15 days. You can get a return reduction only for the part of the basic fare (Joshaken) by 10%.
So as above poster suggests, you might think about buying a railpass if you're tempted to travel by train as cheaply as possible. But, considering you are arriving and leaving on long haul international flights, and will be traveling with large luggage, I wouldn't recommend long train trips in Japan.
Instead, I'd say you should fly. You might find cheap flights between Tokyo Haneda and Oita on ANA site. Prices available mostly from 15,000yen each way, but if you're lucky, you may find a 12,000yen one way ticket for the sector. You can book a seat online, no matter where you may be.
http://www.ana.co.jp/eng/index.htmlIn Japan, domestic flight tickets start to be on sale two months prior to the boarding date. So you should keep an eye on the site in May.
Alternatively, you can fly between Haneda and Fukuoka with Skymark, and catch a bus between Fukuoka and Beppu, as suggested above.
http://www.skymark.co.jp/en/index.htmlPrices from 12,000yen each way. But a bargain price of 10,000 may be available if you try to snatch it as soon as they start to sell two months before.
Anyway, take a look at the sites.
The limousine bus connects between Narita and Haneda for 3,000yen each way and takes 75minutes.
Hope this helps.