If each bitcoin had to be spent 300 times if equal to 1% of the internet economy (at time of article)... is that like saying 1 bitcoin would be worth 300 dollars?

No. You have to take the velocity of money into account.
Suppose I buy something from Amazon with bitcoins. Amazon then pays their employees with those coins. Those employees buy other stuff with their wages. The same money goes round and round in the economy. This is called the
velocity of money.
Based on the 2006 figures that I quoted above, 1% of the US internet economy is around 1.5 billion dollars. If bitcoins were being used for that, it would equal $300 of transactions per bitcoin (1.5 billion divided by 5 million).
If each bitcoin was spent 300 times during the year to support that amount of trade, then each bitcoin would be worth $1. If each bitcoin was spent only 30 times per year, then each bitcoin would be worth $10.
If we look at
bitcoinwatch.com we see that over the past 24 hours, 108641 out of 5450000 coins were transacted. That's 2% of all coins changing hands per day, meaning that on average each coin changes hands just over 7 times per year.
But that's not quite the figure we want. It doesn't reflect purchases of tangible goods (it could be just people moving coins from one wallet to another, or back and forth between their wallet and MtGox). Also, it doesn't include people who make purchases using the shopping cart interfaces of MtGox, MyBitcoin, etc. In fact the velocity of money is almost impossible to measure, and most economists don't even try. On the other hand, the value of a bitcoin (or a dollar) is easy to observe, so that's what people measure.