So I should install an NTE Faceplate filter like this at the BT master socket.
That's the type, although be careful because some of those faceplate filters do not have extra terminals on the back for an unfiltered connection. In other words, once it's installed you can connect your DSL modem only by way of the jack on the front, not by running an extension cable from the rear terminals. To do the latter, you need a unit which has unfiltered terminals as well, for example:
On the back of that unit, terminals 2 - 5 are the usual ones for wiring to a telephone extension and are filtered; terminals A & B connect directly to the incoming line and would be used to run an unfiltered extension to wherever you want your modem to be located.
Then I should run a telephone cable (Phone would be going to the second room) from the NTE to the office. One end plugged in the filter and the other directly into the modem?
For fixed wiring you want the CW1308 cable that Bernard mentioned. It's the standard internal telephone cable with solid cores and paired: blue/white, orange/white, green/white etc. Connections at the filter at the NTE are to the terminals as described above, then you can fit jacks in the office into which the modem and phone can be connected.
Do I need to plug in a DSL filter before plugging in the modem?
The replacement unit you linked to or the one I referenced above is
the filter. You connect the modem to either the ADSL jack on the front, or by way of your extension wiring to the A/B terminals; you connect telephones to the voice jack on the front, or by way of extension wiring to the 2/3/4/5 terminals which are the filtered output. No other filters are then necessary.
I think the cable distance from master socket to modem would be about 30 meters
. Would you suggest running telephone cable or going for ethernet option? I've got switch so either way I'll have to just buy the cables.
Either option would be fine. Unless you happen to live almost top of your telephone exchange, an extra 100 feet or so of telephone cable against the hundreds (or thousands) of yards already running to your house isn't going to make a whole lot of difference to your ADSL so long as it's installed and connected properly.
I would let other considerations dictate which course is the better one to suit your needs, such as if it's a combination modem/wireless router and one location or the other is better for the wireless side, or whether it's more convenient to have the modem/router in the office than using space where the NTE is, etc.