What is the right term for students admitted to the school in previous years, if you are also a student? Do you call them "elder students"?

Solution 1:

The word used in my experience is upperclassman. The dictionary definition is a student of the junior or senior year of either high school or college, but for new first-years, it is often extended to sophomores, juniors, and seniors.

Note that this word is not (politically correct) gender neutral, and so may be phased out in the next twenty years or so.

Solution 2:

x "year(s) ahead of" me
y "year(s) behind" me

He is fond of casually dropping the names of his drama school contemporaries. Ewan McGregor, Joseph Fiennes ('We would potter next door to see Ralph playing leads at the RSC'), and more long-term friends such as Dominic West, whom he knows from his Eton days. 'Dom was a couple of years ahead of me,' Lewis recalls.

from The Telegraph

President Obama was four years ahead of me at Harvard law school....

quoted in The Atlantic

That reminds me: Lee Harvey Oswald, a year or two behind me in school, lived in Covington in those days.

from the New York Times

Prince Harry had started at Sandhurst in May 2005 and was a year ahead of his older brother.

from the Evening Standard

variant:

My favorite Stanford summer was 1979, when a woman engineer in the class ahead of me helped me get a summer job at Chase Manhattan Bank in NYC.

Stanford University website