It's true that t'other clearly means 'the other.' You can see it more clearly in the phrase 'one or t'other,' which was more common in the 19th century than today.
However, I think in the OP, Trollope may be playfully using 't'other' as an adjective, following the indefinite article 'a' (which does not mean 'on' or 'at'). A big school, an old school, a t'other school. What kind of school? A t'other school.
If my guess is right, then in the system he describes, a t'other school could mean something like 'a second-choice school' or 'a second-rank school' as the kind of school often chosen for the youngest boys, when the quality doesn't really matter yet as much, to be followed by the best schools, Eton and Oxford, as the boy grows older.