It is not possible to automatically execute a user or project tool on project load. But you could use not really good workaround.
It is possible to execute a macro on file open. You could write a macro which executes a user tool when a file with a fixed name is opened. I have never tried it, but I think it is not possible to run the macro on opening of the project file itself. But when you specify in the macro to execute the user tool on opening a file with a fixed name and this file is always automatically opened on project load because it was open on last project close you could get what you want. The macro executed on every file open should have a code like this
RunTool "Case sensitive name of tool"
I have also another idea - not tested. It is possible to execute ctags on every project open as you already know. But I suppose that UltraEdit does not check if really ctags.exe is executed. At Project - Ctag Options you can specify the full name of ctags.exe, but you can specify also any other EXE or perhaps even a batch file. And UltraEdit surely does not check the "ctags" parameters passed to "ctags.exe". So you can specify in the general ctags option or in the project ctags options also a different parameter string useful for what your tool needs.