How to analyze a display?
This looks like a 16-character 14-segment plus decimel point and comma (16 segments total) multiplexed vacuum fluorescent display. There appears to be a high voltage anode supply located in the lower left corner (orange transformer coil and nearby circuitry). With only eight wires entering the board at J12, there must be some other part(s) we don't see that perform the multiplex function to accept segment data and clock it into serial shift registers that drive individual display segments. I have included a link below to one such integrated circuit. Note these were available in very small surface mount packages and its presence may not be immediately obvious. The multiplexing circuit is typically one line for each character and one line for each segment or a total of 32 lines. That also happens to be the number of lines that appear to connect the display to the circuit board.

At a minimum, to interface with the display driver, there should be a clock line, a data/command line, a reset line, power and ground. Those signals are probably to be found on the eight wires connected to your board. The multiplexing driver integrated circuit is probably mounted under the vacuum fluorescent display. You will need to figure out the serial data format (ASCII, EBCDIC, Klingon, or whatever), the clocking speed, and how to generate the serial data and the clock with your Arduino. The most significant bit of an 8-bit data stream typically signifies whether the next seven bits are to be interpreted as a command or data. I would start sending 7-bit ASCII streams with no parity to see how the display responds. Also read through the command descriptions on the datasheet to see if there is any sort of setup required before characters can be stored and/or displayed. There is a 16 page PDF data sheet on the OKI Semiconductor MSC1937-01 display driver that you can download from http://www.datasheetarchive.com/14%20segment%20display-datasheet.html# that may be of some help. It is near the bottom of that web page.