As long as you do not wish to program the "Bare Bones Arduino" board, this should work just fine. You can connect it to the "PRG" pinout on the board making sure you do not
make any connection to pin 3, the reset line.
You mention a USB to Serial adapter - Similar
to the one you illustrate. Again, if you do not need to arrange the automatic reset for program downloading, all you will need to do is to identify
the TXD and RXD connections (as well as ground) on it and figure out whether they need to swap over or not as they connect to the Bare Bones board.
In fact you can
in this fashion quite easily use it for downloading sketches as well, notwithstanding the comments on the description page, by learning the right timing to press and release the reset button about one second after the "Binary sketch size" message appears in the IDE.
The USB to Serial module you illustrate (lacking in any useful description of its pin allocations) may or may not as it stands be able to perform the automatic reset for sketch downloading; it will require the 0.1µF capacitor in series to the reset on the Bare Bones board if it correctly
brings out the DTR or RTS line to its sixth pin; and
some fancy PCB track modification as well if it is one of the "bodgie" ones.