How to deal 1000 in destruction to honor structure in last shelter?

Your APC's destruction points are shown in your APC list - 500 will steal a tile.

You will need to hit an honour structure repeatedly until you have accumulated 1000 DP. It need not be all in one hit.

Death Rider (hero) raises destruction, so does the Eden weekly package (for cash), and anything that raises the strength of an APC will raise the destruction per hit.