Welcome to the Bunnings Workshop community @hajayne. It's wonderful to have you join us and many thanks for your question on restoring a lawn.
You’re pretty much right on the money there. Aerating the lawn first is a great idea. It’ll help relieve compaction and improve water and nutrient penetration. After that, applying a quality topdressing and underlay mix will help level out the surface and provide fresh material for the grass to grow through.
For the bare patches, you can either reseed or re-turf, depending on how quickly you’d like to see results. Re-turfing will give you instant coverage, while reseeding will take a bit longer but is more cost-effective. Here are a couple of helpful guides: How to grow grass from seed and How to lay turf.
Manual aerators work perfectly well for smaller lawns like yours. If the area isn’t too large, you’ll get good results using a manual corer or even a garden fork to open up the soil. Mechanical aerators are really only worth considering for much bigger areas.
With some consistent watering after topdressing and feeding, your Sir Walter should bounce back beautifully. It sounds like you’re on the right track.