I am using following code to connect QMenu to QPushButton. When button is clicked a pull-down menu with multiple sub-menu's items is shown.

button=QPushButton()
button.setText("Press Me")

font=QtGui.QFont()
button.setFont(font)
button.setSizePolicy(ToolButtonSizePolicy)

button.setPopupMode(QtGui.QToolButton.InstantPopup)
menu=QtGui.QMenu()
button.setMenu(menu)

menuItem1=menu.addAction('Menu Item1')
menuItem2=menu.addAction('Menu Item2') 

Now depending on a condition I would like to customize QPushButton display by giving it a text and background color. The following line of code (which is supposed to change background color) has no effect on QPushButton connected to QMenu.

button.setStyleSheet('QPushButton {background-color: #A3C1DA}')

I would like to know how to change the background color of QPushButton as well as button's text's color.

Apart from some inconsistencies with your code example setting the background color and text color of a QPushButton works just fine with:

setStyleSheet('QPushButton {background-color: #A3C1DA; color: red;}')

Example (using PySide):

from PySide import QtGui

app = QtGui.QApplication([])

button = QtGui.QPushButton()
button.setStyleSheet('QPushButton {background-color: #A3C1DA; color: red;}')
button.setText('Press Me')
menu = QtGui.QMenu()
menuItem1 = menu.addAction('Menu Item1')
menuItem2 = menu.addAction('Menu Item2')

button.setMenu(menu)
button.show()

app.exec_()

results in: