use sram as logic analyzer?
What you describe is more of a data recorder than an logic analyzer.  A logic analyzer includes the ability to generate a trigger based on the input data.  It stops storing when the trigger sequence and/or pattern is detected so that you can examine the stored data.

Also, you would need to add an external register between the outside world and the SRAM that is clocked by the same clock as the SRAM so that the data is stable as it is being written.

With some refinements, what you describe is possible and you would certainly learn a lot implementing it.  To save time, you could perhaps even use a low cost FPGA development board that has some built in memory as a starting point.

However, if you just need a tool to use now, it's probably best to get the cheapest USB based analyzer with enough memory and speed that you can find.