More specifically, if normal wine doesn't work(which I was able to get it to work, best off trying 32-bit), all hope is not lost, as this sounds like a perfect candidate for the Wine extension/add-on'esque program, 'Wine-Hacks,' it prompts the user with a walk-through for installing various pieces of software that meet particular requirements/etc...(this is a very easy avenue, if traditional wine configuration doesn't work). This is an interesting predicament, and its important to remember that linux in general gives the user a far greater platform to customize, and tweak the various processes/threads/etc on all levels... However, a large amount of reading typically comes from it. On a lower-level perspective one could, if the game is utilizing interprocess communication, tweak the speeds on the sockets('vmstat'/'top'/'ps'/'ss' commands are but a small arsenal of what linux offers) themselves, and although it will not have as small a learning curve/ease of use, but these tools are a start for analyzing process-level activity/memory-mgmt.
But looking up at your image, and having downloaded the app myself, gaining an understanding as to how the application works, and the fact it's reading bits of memory, and scanning memory addresses/etc... it eludes it is very doing just this, tweaking the memory/interprocess communication on a low-level(however going this route you'd be creating your own hackish-version of the 'Cheat-Engine' it could very easily be done with a gdb(gcc-debugger, a developer/security tool as doing these types of tasks are a common part of a developer's job to better gauge how the application works/its overall performance., acts similarly in a more limited scope). You'd just be using a different tool to control the memory/sequencing of socket-level communication. No less, the Wine-scenario will work, and I tried it briefly(however, I don't know what the targeted game your playing is, lol!!).
To conclude, so that we're clear, one way would to re-nice the processes/process threads(tasks) in Linux/and locating the various memory addresses, and bits used in the interprocess/socket communication using utilities/or debugger(already mentioned). Then another more basic way that would utilize the logic of the 'Cheat-engine,' itself in a non-Windows environment, through installing it with Wine, as Wine runs the application, and manages it in processes/threads within so it is as if the program is being ran in Windows itself. The Wine program is in all Ubuntu-specific distros I have seen, "'apt-get'/'aptitude'" it to install Wine). Additionally, another tool for debugging/tweaking flash-based games/etc... is that of Flashbug(a firebug variant, just as Greasemonkey is used in Mozilla)Good luck & cheers!!