Bypass capacitor placement in Eagle
The auto router is a great tool that can replace hours of manual tedium, but it's not a do-all solution.  There is only so much you can explain to the auto router, and therefore have it make the right tradeoffs for you.

Carefully placing bypass caps on the opposite side of the FPGA and carefully routing the tracks to those bypass caps and the power/ground pins is something you're going to have to do manually.  That's the state of the technology today.

My normal workflow in routing a board with Eagle actually starts with giving thought and attention to placement.  Look at the airwires carefully when placing parts, and think about routing at that time.  Then when it's time to start routing, I start by doing the few critical areas manually.  These usually include the high current loops inside switching power supplies, the crystal connections to microcontrollers, and the bypass caps.  If using a ground plane, I manually connect most ground pins to the ground plane with their own via immdiately by the pad.  After that I make sure the ground around important parts is routed correctly, and only then see what the auto router can do.

Always save a copy of the board before running the auto router.  The first few times I use none or only a few optimization passes.  The purpose is to see where the problem areas are and what kind of trouble the auto router will get itself into.  Then I manually route a little in those areas and try again.  After a few passes of this I can see that the auto router looks like it will do a reasonable job on the rest, so I run it with 8 optimization passes.

It's important to tweak the auto router parameters carefully for the type of board you are doing.  This also means changing the parameters over the 9 passes.  In the first pass, I want to maximize the chance it will find a solution, regardless of how messy that might be.  I therefore give a reasonably strong preference to vertical/horizontal orientation of the top and bottom layers and make the via cost relatively low.  In subsequent optimization passes, I remove the orientation restrictions, make via costs increasingly higher, and a few other tweaks.  For the last few passes, I make the via cost a little lower again, favor the outer layers more, and keep making the polygon cost higher and higher.  I use different setups for 2 and 4 layer boards.  On a 2 layer board I usually try to keep the bottom layer a ground plane and break it up only for short "jumpers" only.  On a 4 layer board, the tradeoff are different, so you have to set the various auto router costs differently.

Even after all that, you go back and look at what the auto router did and possibly do a little cleanup.  Even with the hugging parameter set to 0, Eagle will still often lump vias together.  That's usually not good on a 2 layer board where anything on the bottom layer breaks up the ground plane.  For a plane, it is better to have more smaller islands than fewer larger ones.  You want to minimize the maximum dimension of any island.  That means separating vias and their traces from adjacent vias and their traces so that the plane pour flows around them.

Anyway, the point is that if you think you can just throw your design at the auto router and be done with it, then you're in trouble.  The auto router is a great tool, but you have to understand it and work with it, not just delegate everything to it.