The issue is not really, whether the Xonar or similar can handle the impedance, it's more a power issue. I have run my MDR-V700's (which are 24-Ohm, and likely, very similar is spec to your MDR-1A's) directly, off of the Sound Blaster Live!, Audigy, Audigy2, X-Fi, and I always, assumed they weren't very good headphones (as everyone kept trying to make me believe) because they didn't sound very good off of a soundcard. In fact for years I ran a pair of Skullcandy Ink'd Smokin' Buds (IEM's) directly, off the soundcards because they had so much more power and bass. But then their power handling was a mere 73mW maximum or so so the paltry output of a soundcard wasn't so paltry in that case. However, my V700's have a power-handling of between 1W-3W depending on where you searched on the internet.
Skip ahead several years, and I've since began using things such as stereo receivers to power my headphones, while outputting from my soundcards to the receiver. This was already, a huge upgrade from running directly off the cards because the receiver had more power handling. Then skip ahead even further, I've upgraded from an X-Fi PCI to a M-Audio Delta 410 PCI, which was a night and day difference in audio quality, alone. Now, skip ahead even more, and I've pulled out my Dads old Yamaha pre-amplifier, which has its own headphone stage. WOW! The headphone stage on this piece of 80's vintage quip was very powerful, the likes of several hundred milliwatts--IIRC it put me in the realm of around 600mW given my headphones' 24-Ohm load, and the internal IC's given input voltage. Suddenly, my headphones sounded like something entirely new, with much more power, and much warmer than the anemic headphone outputs of the receivers I was using.
In-between all that, I had gone to CanJam in San Francisco and tried my Sony's on several amplifiers. The best experience I had was on the Schiit Magni 3 headphone amp, which put me closer to 2W! I compared several popular headphones, which were floating around the event from Sennheiser and Beyer, etc. and didn't like the sound of any of them compared to my Sony's. (everyone hears differently, so there's going to be times when nobody else can tell you what's better) I've since been trying to save up enough to buy one, but just haven't made it happen, yet.
In your case, if $100 is your maximum budget, you might spring for a Schiit Fulla 2, but you'll exceed $100 with taxes and shipping. If you want to try something to just get you going, I'd look around on craigslist for a modern receiver that has its own headphone jack, can use an S/PDIF input or even USB input so you can just use it as a DAC/AMP combo. This is actually, what I had to do because the Yamaha pre-amp fried one channel so I'm using a modern Sony receiver as a DAC/AMP. Sounds good for now.
