The solo in Rimsky Korsakov's Russian Easter Overture is one of many for the second trombonist, along with those from Sheherazade, Tchaikovsky's Symphony No. 3 and Mozart's Requiem. Presumably the composers were writing for orchestras using an alto trombone on the top part, and wanted the richer sound of the tenor trombone on these solos.
The Russian Easter Overture is based on themes from the musical tradition of the Russian Orthodox church, with the tenor trombone solo representing the medieval chant of a priest. The composer marks the solo "a piena voce" - at full voice; thus, interpret the solo as a vocalist would.
As with any "wide-open" solo, you must know the piece, style, tempo, breathing spots and standard interpretation well before the first rehearsal. In other words, know it by heart and listen to recordings! If you do not have access to the actual part, it is in Volume 1 of the International Music's series of excerpts, and contains Keith Brown's excellent editorial suggestions.
The solo is written with little indication of dynamics or expression, so most trombonists play it at a comfortable mezzo-forte with a broad, rich tone. I find it easiest to let the line itself dictate the interpretation: crescendo as it ascends, decrescendo as it descends, and taper the volume at the end of the phrases. Traditionally, the repeated notes in the 8th measure are played softer, with a long cresendo/decrescendo in measures 11 through 12. The player needs a big breath between the first two C's in measure 11: the last note is held underneath a violin cadenza, and sometimes the Concertmaster takes quite a bit of time to finish!
The traditional interpretation is given in the International excerpt book - you are safe to use those. The solo is accompanied only by soft, sustained strings so intonation is critical. I suggest using a tuner before the first rehearsal just to lock in the correct positions. I have also found that using a small amount of vibrato at the phrase endings helps sustain the tone. Above all, be musical and enjoy the solo!