Spurs
 
The best way to get to White Hart Lane from Kings Cross is to jump on the Victoria Line to Seven Sisters, from where you jump on a train to White Hart Lane station. There are normally additional trains running between Severn Sisters before and after the game. If you hit it right the entire journey from Kings Cross will only take about 20 minutes. Alternatively it's a 25 minute walk from Seven Sisters to the ground, with a couple of pubs on the way that you should be able get in. Whatever you do you'll need a One Day Travelcard for zones 1-4 unless you've got an Oyster Card or registered contactless card.

Anyone daft enough to drive down will find that there is no street parking whatsoever anywhere near the ground. You'll therefore have to either park some distance away or pay a lot of money in a private car park.
 
Pubs
The pubs closest to the ground are strictly home fans only, and many others are very partisan and so you might want to give them a miss. The best options are therefore probably:
 - Gilpin's Bell (Wetherspoons, which is a 10 minute walk north of the ground along High Road)
 - the Elmhurst, which is where any Yorkshire Mags going to Spurs in recent seasons have tended to go; get off at Bruce Grove if you catch the train from Seven Sisters
 - Harringey Irish Club, very close to White Hart Lane station
The above, as well as many others, are all marked on the pub map below.

Alternatively you could just drink around Kings Cross and catch a late tube up. If so then be aware that you'll need to set off around 2 to make kick-off.

For more info on getting there, pubs, etc check out:
Pub map
Kings Cross pub map
Internet Ground Guide
London Transport
Official Spurs site
Top Spurs (unofficial site)