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.