Does j2me program can run at the phone backgroud?
I appreciate your asking the questions here but you may want to spend a bit of time doing a google search and see if these questions are already answered.
If you need to know how to do this, that is another question and needs to be done in more detail.
Run application at the phone background is depend on devices firmware or you can say device specific.
For Nokia series40 devices you can run in background without using 'getDisplay(Displayable)' method. But actually it is running without any display screen and any other application will not run until it is distroyed.
As Ron said , you will get lot of discussion by googling.