actually the 2zz motor has a little more issues, due to it's high performance nature. as such, 2zz owners tend to "enjoy" LIFT (powerband from 6200 - to redline) a bit too much.
The LIFT bolts do need to be properly cared for as well.
Other than the power advantage, 1zz motors do have better fuel mileage and a slight torque advantage over the 2zzs - of course until LIFT kicks in.
In your case, it may just a bad motor which is not a common issue.
Did you get your car used or new?
If used, the prior owner may not have cared for it properly.