By plane
You can reach Atami from one of Tokyo's two airports. From Narita Airport, take the Narita Express train to Shinagawa, then transfer to either a Shinkansen Kodama train or a Tokaido Line local train. It is 2 hours via shinkansen connection (�7100) and 3 hours via local connection (�4900).
From Haneda Airport, take a Keihin Kyuko line train to Shinagawa (�400) for connection to the Shinkansen (1 1/4 hours, �4500 total). Alternatively you can take the Keikyu Line from Haneda to Yokohama via Keikyu-Kamata station (�470) and change at Yokohama for the local JR Tokaido line (2 hours, �1750 total).
By train
Atami is on the Tokaido Shinkansen main line, just 50 minutes from Tokyo (�3570 for an unreserved seat). All Kodama trains stop at Atami, and a select number of Hikari stop there as well.
Kodama trains make the run to Atami from Shizuoka in 40 minutes (�2960 unreserved seat) and Hamamatsu in 70 minutes (�4930 unreserved seat).
Atami is also a stop on the regular Tokaido Line which runs from Tokyo (2 hours to Atami, �1890), Shizuoka (80 minutes, �1280) and Hamamatsu (2 3/4 hours, �2520).
With the exception of the Keihin Kyuko line, all train rides listed above are free with the Japan Rail Pass