DIED
"On 26th May 1851, at his station, Murrumbidgee, John Rochfort Rae Esquire, after a short illness."
If there was a church burial record, it is unlikely to give you any more information than you have from the newspaper.
Yes, he would have been buried on the station. There was perhaps a travelling clergyman, who may have said prayers at the grave site on his visit to the station, which could have been many months later. If you find out if there was a travelling clergyman, there may be note of any such event in his papers / diary.
"Uardry" was initially established in 1840 by John Rochfort Rae, with his partner Henry Angel arriving in 1844. The property was originally known as "Wardry".
You might find more information from the Wagga Wagga Historical Society.