To inflate popupMenu from a button onClick, use the following code.
btn = (Button) findViewById(R.id.btn);   
btn.setOnClickListener(new View.OnClickListener() {
        @Override
        public void onClick(View v) {
            PopupMenu popup = new PopupMenu(MainActivity.this, v);
            popup.getMenuInflater().inflate(R.menu.pop_up, popup.getMenu());
            popup.setOnMenuItemClickListener(new PopupMenu.OnMenuItemClickListener() {
                public boolean onMenuItemClick(MenuItem item) {
                    Toast.makeText(MainActivity.this, "Some Text" + item.getTitle(), Toast.LENGTH_SHORT).show();
                    return true;
                }
            });
            popup.show();//showing popup menu
        }
    });
EDIT
To style the popupMenu, add the following style.
<style name="PopupMenu" parent="@android:style/Widget.PopupMenu">
        <item name="android:popupBackground">#ffffff</item>
    </style>
I noticed you also want to add icons next to your text. It is possible to add icons in popupMenu. However it is a better approach to use popup Window instead. Here is a sample code:
PopupWindow mypopupWindow;
setPopUpWindow();  
     btn=(Button)findViewById(R.id.btn);  
     btn.setOnClickListener(new View.OnClickListener() {  
       @Override  
       public void onClick(View v) {              
           mypopupWindow.showAsDropDown(v,-153,0);  
           //showAsDropDown(below which view you want to show as dropdown,horizontal position, vertical position)  
         }  
       }  
     });  
   }  
   private void setPopUpWindow() {  
     LayoutInflater inflater = (LayoutInflater)  
         getApplicationContext().getSystemService(Context.LAYOUT_INFLATER_SERVICE);  
     view = inflater.inflate(R.layout.popup, null);  
     Start=(RelativeLayout)view.findViewById(R.id.start_btn);  
     Pause=(RelativeLayout)view.findViewById(R.id.pause_btn);  
     Stop=(RelativeLayout)view.findViewById(R.id.stop_btn);
  mypopupWindow = new PopupWindow(view,300, RelativeLayout.LayoutParams.WRAP_CONTENT, true);
popup Layout
<?xml version="1.0" encoding="utf-8"?>  
 <LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"  
   android:orientation="vertical" android:layout_width="wrap_content"  
   android:background="@drawable/whitedrawable"  
   android:paddingRight="0dp"  
   android:layout_marginRight="0dp"  
   android:layout_height="wrap_content">  
   <RelativeLayout  
     android:id="@+id/btn1"  
     android:layout_width="wrap_content"  
     android:layout_height="wrap_content">  
     <ImageView  
       android:layout_centerHorizontal="true"  
       android:layout_centerVertical="true"  
       android:layout_alignParentLeft="true"  
       android:layout_width="wrap_content"  
       android:layout_height="wrap_content"  
       android:src="@drawable/startimg"  
       android:id="@+id/startimg"  
       android:paddingLeft="10dp"  
       android:paddingRight="10dp"  
       android:paddingTop="5dp"  
       android:paddingBottom="5dp"  
       />  
     <TextView  
       android:layout_centerHorizontal="true"  
       android:layout_centerVertical="true"  
       android:layout_width="wrap_content"  
       android:layout_height="wrap_content"  
       android:paddingRight="0dp"  
       android:text="Start"  
       android:layout_toRightOf="@+id/startimg"  
       />  
 <!-- Continue for other items-->
The whitedrawable can be used to set a background of your choice. You can use 9patch to get the shadow and rounded corners for the background.
To dismiss the popupWindow, use the following code:
mypopupWindow.getContentView().setOnClickListener(new View.OnClickListener() {
    @Override
    public void onClick(View v) {
        mypopupWindow.dismiss();
    }
});
To dismiss using the back button, use:
@Override
public void onBackPressed() {
    if(mypopupWindow.isShowing()) {
        mypopupWindow.dismiss();
        return;
    }
    super.onBackPressed();
}