When you were born, and without Googling or Wiki-ing....

[Dr. Cynic]

Let's see... February 11, 1988...

Easy(er):

President of the USA - Ronald Reagan
Vice President of the USA - George Bush
Monarch of the UK and other realms of the Commonwealth - Elizabeth II
Prime Minister of the UK - Margaret Thatcher
President or Premier of Russia or the Soviet Union - Gorby
President of France - Francois Mitterand
Pope of the Catholic Church - John Paul II

Somewhat more difficult:

Speaker of the US House of Representatives - Tip O'Neill
Majority Leader of the US Senate - Bob Dole, I think
Prime Minister of Canada - Brian Mulroney
Prime Minister of Australia - Bob Hawke

Considerably trickier:

Governor of Massachusetts - Mike Dukakis
Governor of Illinois - I don't honestly know...
UK Chancellor of the Exchequer - It was either Nigel Lawson or John Major... Lawson, I believe...
UK Leader of the Opposition - Neil Kinnock
Most recent Best Picture Oscar winner - Since the 1987 Academy Awards were held after I was that means that Platoon was still on top.
Most recent Nobel Peace Prize laureate - Got me, couldn't even guess...

Very tough:

Senior U.S. Senator of Tennessee - Jim Sasser
Junior U.S. Senator of Vermont - No idea. Pat Leahy maybe?
Leader of the Scottish National Party - Gordon Wilson (Thanks Prime Minister Forever UK!)
Most recent Best Supporting Actor Oscar winner - I believe it was either Michael Caine or Denholm Elliott who were still reigning from '86... I'm not sure... I am a big fan of both guys and I know they were both nominated, just not sure who won.
Most recent Nobel laureate in Physics - Search me...