First generation cars always have the bugs that aren't worked out until later iterations. I believe the first gen MDX made the list of top least reliable cars due to transmission failures.

They are ok but they were at the top of the least reliable cars once in Scottys list on average falling apart at 190k miles. The transmissions are weak. 

I have one so I may be biased but I’ll share my experience. I have about 183,000 miles on mine. I got it at 164,000 miles. It needed tires when I bought it and brakes soon after. Since then? Oil changes and that’s about it. I drove it in the snow, it handled amazingly compared to my Nissan Sentra. I’m very pleased with mine, and I’ve had 0 issues with the transmission, but from my research it’s the 2003 MDX with the most transmission failure, mine is a 2004 so I don’t know maybe I’m lucky?

We bought a 2003 MDX when it was about 4 years ago with about 70,000 miles on it.  It now has 144,000.  I have maintained it according to Honda's maintenance schedule.  We had to replace and alternator, power steering seals, and one of the power windows broke.  I also had to replace the things to keep the hood up.  It has a minor leak from the oil pan.  I have had to repair the AC vents and the CD player doesn't work. For a 17 year old car, that has had hard miles with lots of short trips since we owned it I am very impressed.  It runs and drives like new. 

So how reliable is this car.  Very.  Not like a German or American car and certainly better than a Korean car.  No offense intended to anyone.  We traded in our very unreliable and unrepairable Mercedes E class for this car when we bought it.

But a car is a machine and when they get old, a lot depends on how they were maintained.  I change may ATF according to the schedule using Honda fluid. That might make a difference. I can't imagine that you would have any engine problems with this car.  But I expect you will have some power steering issues.  The CD changer probably won't work, the vents will be broken and need repair.  You will find vent tabs on Amazon for this car that are great. 

I love this car.  It may be first generations, but is really well done.  It has all the perks without the complications of newer cars.  If you can find one that has been well maintained, I recommend it.  However I want to warn you that gas mileage is bad.  It is a large and very powerful car.  But it was made at a time where things were still very simple making it repairable and reliable.