Sometimes the only way to find out is to ask.

However, if you can find out WHEN a priest was ordained you can often find out whether or not further investigation is necessary:

Any priest ordained before 1968 is undoubtedly valid.

Any priest ordained before 1969 is valid assuming that the consecrating bishop was consecrated in the old (pre-1968) rite.

I have actually had a very difficult time finding out exactly when the NREC and the NRPO became obligatory.  So far have only been able to narrow it down to the year.  If anyone can provide the actual date, that would be helpful.

If a priest was ordained in 1969 or later, more information would be needed.  It's possible that he was ordained in the traditional rite even if it was 1969, presuming again that the consecrating bishop received the episcopacy before 1968.

After 1969, the NRPO was obligatory, so any bishop who was following the rules would have been using it.  That naturally casts doubt on a lot of these priests' orders for some traditional Catholics, and even moreso when New Rite bishops are the ones doing the consecrating.

Obviously, the traditional orders are a completely different story.  

As to finding out which bishop ordained him, that would be generally easy enough to find out if you know when they were ordained, and where (meaning, which diocese).  You can use catholic-hierarchy.org to find out who the ordinary would have been at the time, and also find out when he was consecrated.

I'm trying to find out the bishop who ordained a priest in 1992.  I don't see how to find this info on the website you provided.  In my internet research I found out that the priest was ordained in Rome.  

In answer to your question about how to use the catholic-hierarchy.org site, you would go to the toolbar at the top and mouse over "dioceses" and select "all."  From there you would find the diocese where the ordination took place, and click on it.

From there, it will give you the history of the presiding bishop(s) in the diocese.  If, for example, the ordination took place in 1985, you could see who the bishop was at that time.  By clicking on that bishop's name, you can find out when he received the episcopacy.  

Now, the ordinary of at the time isn't necessarily the person who would have ordained the priest, but there is a list at the bottom of the page for a particular diocese of "associated bishops."  Auxillary, co-adjutators and such.  These are simply methods to narrow it down.

In all honesty, in investigations of my own, they've almost all ended in finding out that the bishop performing the ordination was consecrated in the new rite.  From there, it's pointless discerning what rite the priest was ordained in since the bishop performing the rite has doubts surrounding his episcopacy.