Japan Railways, JR does not classify their trains as 1st class and 2nd class but Green car and ordinary car. Therefore you want to travel on Green car=1st class in Japan, you have to buy the Japan Rail Pass for Green car.
Most of the Shinkansen and limited expresses are equipped with Green Car. The difference between Green car and ordinary car is almost same as European 1st class and 2nd class, except that Green cars are compulsory seat reservations. Which means you can not sit down on any unoccupied seats if you did not make any seat reservation in advance. Actually ordinary cars are also divided into reserved cars and non-reserved cars.
Japan Rail Pass holders are able to ride the reserved cars without any extra supplement but you have to choose which train to take in advance. Then, you have to go to the ticket office to get the seat reservations.
The busy seasons in Japan are 1) the end of April and beginning of May = Golden Week 2) Middle of August = Obon and 3) End of December and beginning of January = Nenmatsu nenshi.
During these periods, JR runs dozens of extra limited trains which do not run on other periods. Mostly these extra trains will not be listed on third parties websites or applications. Therefore please check at the JR's websites.