How can I use getSystemService in a non-activity class (LocationManager)?

You need to pass your context to your fyl class..
One solution is make a constructor like this for your fyl class:

public class fyl {
 Context mContext;
 public fyl(Context mContext) {
       this.mContext = mContext;
 }

 public Location getLocation() {
       --
       locationManager = (LocationManager)mContext.getSystemService(context);

       --
 }
}

So in your activity class create the object of fyl in onCreate function like this:

package com.atClass.lmt;
import android.app.Activity;
import android.os.Bundle;
import android.widget.TextView;
import android.location.Location;

public class lmt extends Activity {
    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.main);

        fyl lfyl = new fyl(this); //Here the context is passing 

        Location location = lfyl.getLocation();
        String latLongString = lfyl.updateWithNewLocation(location);

        TextView myLocationText = (TextView)findViewById(R.id.myLocationText);
        myLocationText.setText("Your current position is:\n" + latLongString);
    }
}