No, you dont need a reservation for travelling in a regular seat.
You need a reservation for a couchette though - this is not available online but can be made at stations in many European countries (sometimes you would have to go to the international counters).
You should be fine though making a reservation a few days in advance in mid-june.
EDIT: I just checked, you can make the reservation online at https://acpres.rail.cc
but it will be more expensive compared to buying at a ticket desk.