Prints are mirrored in X-axis and inverted in Y-axis direction
For most Cartesian printers, the homing position is at the front-left corner of the build plate. End-stop switches can be at either end of each axis (and even both), but the firmware must be configured accordingly.
A common arrangement is to have end-stop switches at X-min, Y-min and Z-min positions. You will see this on pretty much all budget printers, but things may be different on high-end machines. For the Y-axis on a cartesian machine, this mean placing the end-stop switch at the rear of the printer. A CoreXY machine on the other hand has the Y-min sensor on the front left corner.
So, unless your intentions were otherwise, you have simply got the Y-axis end-stop switch in the wrong position. For your design, it should be at the back of the printer, triggered by the bed in it's most backwards position. You will also need to reverse the direction of the Y-axis stepper motor, do that +Y moves the bed towards the operator (like you have it now).
If you want to have the end-stop switch at the front of the printer for some reason, you will need to re-configure the firmware accordingly - it is an Y-max sensor in that position!