I'm worried that the graphics will put a quick stop to my efforts of playing Fallout 1 & 2

How long do you think it will take me to play both games (time I will spend in the games themselves).
Dude, graphics have nothing to do with the game, it's about whether or not you can handle the turn-based combat and point and click gameplay and whether or not you like reading because most dialogue is not spoken to you.
You will probably mess up the first 2 or 3 runs on fallout 1 so I would say that it would take you a couple of weeks to play it.
And fallout 2, I dunno man, could very well take you a month or so.
You will probably mess up the first playthroughs there as well and have to start over and once you finally have a great character that you're comfortable with you're going to be in maze of confusion.
But if you follow a guide on the internet then I guess you could complete fallout 1 in a couple of days and fallout 2 in 2-3 weeks.