Hi @BeginnerPaul,
Thank you for your question and welcome to the Bunnings Workshop community, it is fantastic to have you with us.
While it is fairly unlikely you'd require building approval, I'd suggest you start by contacting your local council to confirm this. Rules vary from region to region, and it is always better to be safe than sorry. A quick phone call to confirm this can save you a lot of pain further down the line.
It is hard to give definitive answers to this question as structures like arbours are subject to a number of variables that can alter footing design greatly depending on things like the wind rating of the area and the soil quality. If you want a definitive answer, you will need to contact an engineer for their assistance in designing a suitable footing. They will be able to use the Pryda Post Anchor Design Guide to design footings that are suitable for your arbour.
With this being said, footings that use post anchors are generally 300mm in diameter and at least 600mm deep. I would suggest using a high wind anchor such as this Pryda Post Anchor High Wind 450 x 90mm M12 as opposed to the one you've linked to above.
Footings can be dug with a Post Hole Digger which has a 300mm auger that is designed for this exact purpose.
Let me know if you have any further questions.
Jacob
