Being able to play arpeggios (and being able to play them in a moments notice) is one of those basic skills that you should definitely have as a guitar player.
No, you don't have to be able to shred through arpeggios at 275 BPM. However, you should at the very least have the ability to arpeggiate a D#m around the 8th fret of your guitar.
This is not some made up situation. See what happens when you try to target chord notes in a guitar solo (which is what you should be doing if you want to play a solo that sounds 'in' with your progression). Whether its by ear or otherwise, if this is something you are not able to do, then basically you are just noodling around and hoping you run into something that sounds okay (which really is not the most efficient way of doing things).
The best thing you can do is work on being able to use both your ear AND your knowledge of the fret board to play through arpeggios.
As someone who teaches guitar for a living, it has become apparent to me that there aren't a lot of guitarists out there who are able to do this. This is only because most don't know the best way to practice it. Many will think that it will just come to them after a while of playing. While others believe its too difficult to even spend any time on.
It is my job to show people that this just isn't the case. I will demonstrate to you (in the video below) a very easy exercise that you can use to instantly find an arpeggio anywhere on the fretboard. If you start incorporating this in your practice every day, you won't believe how quickly your fretboard knowledge will increase.
This is an exercise I like to call "Intelligent Guitar Hero". It's just like in guitar hero, where its all about hitting the correct note, except you get to learn a skill you can use on a real instrument!
Be aware that this in an exercise that is entirely system-agnostic. It can even be used with my not-so-favourite guitar CAGED system I personally would not choose to do it that way, but it's up to you!
The fun part with this is that you can make it as simple or as challenging as you would like depending on the speed and chords you use. If you are more of a competitive person, share this exercise with your friends and see who can play through it the fastest!
Tommaso Zillio is a professional prog rock/metal guitarist and composer based in Edmonton, AB, Canada.
Tommaso is currently working on an instrumental CD, and an instructional series on fretboard visualization and exotic scales. He is your go-to guy for any and all music theory-related questions.