If 150 is your limit, then that's what it is. I figure 110 for a rod, 30 for a reel, 5 for backing and that leaves.... 5 for a flyline. So much for "a la carte".
Mail order will add at least ten dollars in shipping, so ebay isn't a good option if you order from a couple of different vendors. If you were to order from one of the big box sports shops you could get an outfit for around a hundred bucks. It would be rock bottom in quality, but it will get you out there.
A fresh water reel might last long enough for you to save up for a tougher reel. If you rinse it off, I suspect it would last a season (maybe a year). Those Okumas seem to be good values, but might be way out of your $ range.
As for rods, I would try to get a used one, or one marked down. Yesterday I saw a Sage 8wt (last yrs. bottom of the line) marked down to $107 dollars, so I think that you could find a decent rod in your buget.
You'll spend around 5 bucks on backing. Don't expect it to be free.
I would go the cabellas/basspro outfit route. It will be within your buget, and if you find that you like flyfishing, you will find a way to get into better gear.
Maybe you local shop has some demo gear and stuff lieing around. Can't hurt to ask.
For just a little more.....