What are the benefits of running java 64 bits Minecraft?
I'm wondering if I was to run minecraft in a 32bit or 64bit ubuntu server; would there be a benefit (speed improvements) or something else, or just the same? Downsides?
Your question regarding the benefits of running java 64 bits Minecraft can be answered as - Minecraft is written in Java, so it should "adapt" itself to both 32-bit and 64-bit systems (depending if you have a 32-bit or 64-bit Java runtime).
While Minecraft itself is not optimised for 64-bit, it will take advantage of JVM optimizations for 64-bit processors. Basically, this means more general-purpose registers available (16 in 64-bit against 8 in 32-bit, which means better machine code and less RAM access, improving performance) and larger addressable memory. On the other hand, since pointers take up twice as much space, there is a little extra overhead for using 64-bit.
But you should also think about the operating system, as it will also take advantage of 64-bit improvements. If you have more than 2GB or 3GB of RAM, your operating system should be 64-bit, else it won't access all RAM (or will have a considerable overhead doing so). And if you have a 64-bit operating system, you should also go with a 64-bit Java Runtime for better performance (as other users already commented). By the way, in my opinion, you should always go to 64-bit unless you have a very good reason to avoid it, like compatibility reasons (which is not the case).