I did it some years ago without a guide - I was OK with that. I did arrive very early (this was in September 2012, and back then you could enter the camp without a guide before 10am; maybe it has changed now or they are more strict for August?).
It may be difficult to get there by 8:40 by public transportation. You can google for the bus schedules - that's how I would find them now. Not too hard to find. You'll want a bus to Oświęcim (Polish name for Auschwitz). The bus station in Krakow (assuming you are starting there like most people) is next to the train station. I think it takes about an hour. I bought my ticket to Oświęcim directly at the bus station and confirmed that it stopped at the camp not just the town. (They know tourists use this bus so it's a common question.) In my case, I had to walk about 1/3 of a mile from the road where the bus stopped to the camp parking lot. Some buses or mini-buses will stop directly in the camp parking lot. I wasn't the only tourist on the bus - it was pretty obvious where to get off, because several of us were.
To get back to Krakow, I tried to follow the posted bus schedule at the camp but the bus I was expecting never showed up. A few of us were waiting for it. Eventually we just returned to the road where I had been dropped off in the morning, and we got on the next mini-bus that came by that was headed to Krakow. Not really a big deal. There should be other tourists doing the same thing.
There is train service from Krakow to the town of Oświęcim, too, but you'd need to walk quite a ways from the train station to the camp, as I recall; the bus gets you really close. You can probably estimate how long the walk would take with Google Maps - that's how I would do it.
It may not cost much just to pay for a guided excursion from Krakow to the camp for the day including a tour - might be a lot easier. I'm savvy with public transportation and don't mind a little challenge, so I didn't mind. Depends on your personal preference.