How can I start android application info screen programmatically?

In 2.2 and below, there is no public APIs you can access. But you can still start the InstalledAppDetails activity just as the ManageApplications does. see here

 // utility method used to start sub activity
 private void startApplicationDetailsActivity() {
     // Create intent to start new activity
     Intent intent = new Intent(Intent.ACTION_VIEW);
     intent.setClass(this, InstalledAppDetails.class);
     intent.putExtra(APP_PKG_NAME, mCurrentPkgName);
     // start new activity to display extended information
     startActivityForResult(intent, INSTALLED_APP_DETAILS);
 }

Conclusion: you can start the "application info" screen like this i wrote:

private static final String SCHEME = "package";

private static final String APP_PKG_NAME_21 = "com.android.settings.ApplicationPkgName";

private static final String APP_PKG_NAME_22 = "pkg";

private static final String APP_DETAILS_PACKAGE_NAME = "com.android.settings";

private static final String APP_DETAILS_CLASS_NAME = "com.android.settings.InstalledAppDetails";

public static void showInstalledAppDetails(Context context, String packageName) {
    Intent intent = new Intent();
    final int apiLevel = Build.VERSION.SDK_INT;
    if (apiLevel >= 9) { // above 2.3
        intent.setAction(Settings.ACTION_APPLICATION_DETAILS_SETTINGS);
        Uri uri = Uri.fromParts(SCHEME, packageName, null);
        intent.setData(uri);
    } else { // below 2.3
        final String appPkgName = (apiLevel == 8 ? APP_PKG_NAME_22
                : APP_PKG_NAME_21);
        intent.setAction(Intent.ACTION_VIEW);
        intent.setClassName(APP_DETAILS_PACKAGE_NAME,
                APP_DETAILS_CLASS_NAME);
        intent.putExtra(appPkgName, packageName);
    }
    context.startActivity(intent);
}

From API Level 9 (Android 2.3) you can start an Intent with android.provider.Settings.ACTION_APPLICATION_DETAILS_SETTINGS. Thus:

packageName = "your.package.name.here"

try {
    //Open the specific App Info page:
    Intent intent = new Intent(android.provider.Settings.ACTION_APPLICATION_DETAILS_SETTINGS);
    intent.setData(Uri.parse("package:" + packageName));
    startActivity(intent);

} catch ( ActivityNotFoundException e ) {
    //e.printStackTrace();

    //Open the generic Apps page:
    Intent intent = new Intent(android.provider.Settings.ACTION_MANAGE_APPLICATIONS_SETTINGS);
    startActivity(intent);

}

I used Paolo solution in order to open the Application Details Settings in SDK 23+ when the user turned down the permission request in the past and choose the "Don't ask again" option in the permission request system dialog.

But in this case I used the getPackageName() method directly.

Intent intent = new Intent(android.provider.Settings.ACTION_APPLICATION_DETAILS_SETTINGS);
intent.setData(Uri.parse("package:" + getPackageName()));
startActivity(intent);