the sydney funnel web is nothing to fool with. if you smoke you could light up a cig and blow the smoke to smoke it out like mentioned before. spiders have what are called book lungs. they cant really hold their breath. the air just freely goes through their lungs. once it is out it will probably make its way to the ground a scurry away. as for what you do afterwards is up to you. i wouldnt try to capture it for obvious safety risks. then afterwards you can cover up the hole to prevent the possibility from coming back for a while. funnel webs are crazy toxic and there is no A-V for their toxin. wear gloves and boots with your pants tucked in. its female im assuming since males usually wander around looking for mates. just be carful man. and good luck.
there are a few aerosol sprays that kill spiders instantly too. like the wast killer. you might wanna look into those and they shouldnt harm your plants. just in case though if you use them i would flush out the pot many times.