Yes, I've beaten both the arcade and NES versions of Contra without using the code for extra lives. The trick to getting good at Contra is to devote at least one month to this game, and playing it like a man possessed.
After about a month of dedicated play, you should be so familiar with all the stages that you're seeing them in your sleep. Contra is one of those titles that require quite a bit of memorization but also quick reflexes. The enemy patterns are generally easy to remember, especially in the first couple of Contra titles in the series.
Damn I still haven't been able to do it. Maybe it's something I'll revisit since you're telling me it's beatable. I've had a hard time with that game since I was 8 years old. Almost as much of a hard time as I've had with Metroid the original and super Metroid.
It's definitely beatable. However, you will probably need to sit down and play Contra for a month straight without touching another game. I found that was the best way to get really good at it. The cool thing is that once you've mastered it, you'll be able to go back and beat it again without struggling so much. You'll still need to practice a bit if you haven't gone back to it for years, though.