Oh WOW. So THAT's what you meant by cranking! That makes WAY more sense. Double thumbs up for the youtube vid. That was a stroke of genius.
Sounds like one of your cylinders isn't firing. If it smells like gas I am amazed the car didn't throw a code.
Alright, the car runs well enough after a bit, so timing doesn't seem like it'd be the issue. Fuel/air mix is determined by preset values in the ECU's memory, so that's probably out. If it smells like gas, then you're definitely getting fuel. HMMM.
My first thought is fouled spark plugs. Take your plugs out and look at them. Here's a chart showing what you're looking for. [http://www.ducati.ms/forums/attachments/ducati-motorcycle-chat/74123d1281494359-ngk-mar10a-j-plug-look-okay-you-densosparkplugchart.jpg|"Reading" your plugs] If any are fouled or damaged, replace them. A full set of NGK's shouldn't cost you more than $35.
My second guess would be one of the sensors. OBD-II O2 sensors are self-testing, so it would throw a code if one of those were failing. The next culprit is the Mass Air Flow sensor. this sensor is usually mounted in a rigid tube just after the filter box and before the intake manifold. I used to have a TON of issues with my old jetta when that sensor fouled up with oil from my over-oiled K&N filter. ANYWAY, most auto parts stores sell CRC Mass Airflow Sensor cleaner. its a spray that costs around $5. Personally, I like to clean that sensor as part of regular maintenance every third oil change or so, depending on what my gut says.
The next thing I'd make sure of is that all my injectors were "clicking" correctly. This is the part I love to do where people can see me 'cuz I get all kinds of funny looks. Your local parts store should sell a "diagnostic stethescope." Just like doctors use, only instead of that invariably freezing cold flat plate, it will have a metal rod. When the engine is just starting and exhibiting that issue, touch the tip of that rod to each fuel injector (they'll probably be under those plastic covers on the top of your engine.) What you're listening for is the rapid clicking of that injector opening and closing at high speed. The stethescope should make it really easy to hear if you're touching it directly to the injector. If any of the injectors is clicking more slowly or worse, not at all, then you've got a stuck injector. You can try using an injector cleaner or fuel system cleaner from your parts store and hope for the best. My personal favorite is SeaFoam. It's about $8.00 a bottle. For best results, add it to a low gas tank (about 1/4 tank) and drive around until you're almost out of gas. If that doesn't free any stuck injectors, your next hope is finding a shop that will take out the injector and try manually cleaning it for you. Some shops have really tricked out setups for just these sorts of operations. If all else fails, you'll have to get a replacement, and I really wish I could tell you it will be cheap. Injectors typically run around $80 and up. Try [http://www.rockauto.com|Rock Auto] if you don't have any luck with local parts suppliers. As with any parts, going to the dealer is a last resort, since you will almost always end up paying more than double.
As a part of your due diligence, I would also do a compression test. It will give you a good idea of the health of the engine. Detailing that would take awhile, so [http://www.youtube.com/watch?v=X_tbksFYhl4|here's a good video on the subject.] Personally, I like to remove the fuses that give power to the ignition and fuel system when I do this test. He bypasses those systems in other ways, but I prefer a hard disconnect. Your user manual should tell you which fuses they are.
I'm sure there's something I missed, but this is everything I can think of off the top of my head.
Finally, if you're really gonna commit to keeping this car alive for years to come, [http://www.amazon.com/1998-HONDA-ACCORD-Service-Manual/dp/B004M3II46|You're gonna need one of these.] NOTHING will help you more than the right literature. Those Chilton and Haynes manuals they sell in parts stores are barely worth the paper they're printed on.
And as with any auto repair project, Good Luck!