How to catch a unique constraint error in a PL/SQL block?

EXCEPTION
      WHEN DUP_VAL_ON_INDEX
      THEN
         UPDATE

I suspect the condition you are looking for is DUP_VAL_ON_INDEX

EXCEPTION
    WHEN DUP_VAL_ON_INDEX THEN
        DBMS_OUTPUT.PUT_LINE('OH DEAR. I THINK IT IS TIME TO PANIC!')

I'm sure you have your reasons, but just in case... you should also consider using a "merge" query instead:

begin
    merge into some_table st
    using (select 'some' name, 'values' value from dual) v
    on (st.name=v.name)
    when matched then update set st.value=v.value
    when not matched then insert (name, value) values (v.name, v.value);
end;

(modified the above to be in the begin/end block; obviously you can run it independantly of the procedure too).