- Lying about your achievements is obviously wrong--don't do that.
- Consider that maybe your colleague deserves some of the recognition he's getting.
- Don't worry about your colleague or how he's doing "better than you".
- There's still nothing wrong with marketing yourself a little better as long as you don't compromise your principles.
First, let's talk about your colleague
A colleague of mine is very good at PR and branding himself and does this by communicating to the whole world his successes no matter how small they are. He does this in a very arrogant way, by making it sound like he has achieved this, and often puts an overly positive spin, masking the reality of the situation which is often not ideal.
The trouble is, he is now getting ahead by mingling with important people and is being invited to important events...
Sounds like his career is starting to take off. He's leveraged his accomplishments (real or imaginary) to "climb the corporate ladder" to the point where he is mingling with the important people as you say.
Perhaps he is being arrogant to the point of misrepresenting the facts, as you say, or maybe he's just really good at putting a positive spin on his accomplishments and networking. Without specific examples, it's not easy for me, an internet stranger, to say. If he is indeed "masking the reality of the situation" as you say he is, you already know that that strategy can't work in the long-term. If you continue in a cycle of lying, you're bound to trip over yourself eventually.
But there's also a possibility he might actually have accomplished something here and his newfound recognition is deserved. Keep in mind that you almost certainly don't see everything from your point of view.
Now let's talk about you
In contrast, I like to get work done, and not make a big deal about it. I also like to have a balanced view and do not make situations sound better than they are.
...I am starting to feel like I am being left behind.
If I can offer you a single piece of advice for future success and happiness, it would be this: Don't compare yourself to other people! There will always be someone in your life that you think has it better than you do, and perhaps they don't truly deserve it (or maybe they do). That's not your concern. You'll only make yourself miserable thinking about it. Getting caught in these thought vortexes is not healthy or productive.
It sounds like you get fulfillment out of getting the job done, and getting it right. That's terrific! Focus on doing things right and producing quality output. You may not experience a meteoric rise to rockstardom, but people will notice.
Finally, the answer to your main question
I am starting to feel I need to become more like him, blow my trumpet after every little success, but I really don't want to do so. I find this egotistical.
Can you get ahead in the corporate world without being 'arrogant' and acting with an over inflated ego?
You don't need to be an egotist or a liar to be successful, but there's nothing wrong with learning from what has helped your colleague without compromising your standards. You might need to think about promoting yourself a little more, without lying, of course. For me, this meant learning to focus on the positives a little more.
In morning standup meetings, I used to make the mistake on focusing too much on what went wrong yesterday and how much I was struggling. I found this was eroding management's confidence in my ability to do my work when that really wasn't the issue at all, I was just focusing too much on the negatives, without presenting what I was planning to do to resolve them. So instead, I started saying things like, "I ran into trouble with [X] issue yesterday, but today I'm planning on trying approach [Y] and [Z] to resolve the issue." That gave management insight into the fact that there was a problem, but also that I had a good idea of what to do to fix it. And when I didn't have a good solution, it turned into a discussion where someone would help me find a better approach. This hasn't made me wildly successful, but it has drastically improved my relationship with management and helped us understand each other a little more. And when I finally completed something, management already knew that the work that I put in was quality work, because they were part of it.
For you, it might be a different application, but I think the lesson is the same. There's nothing wrong with "selling yourself" as long as you're telling the truth.