Fairbank's book is by no means bad or even not good, but for your purposes Spence fits much better. Fairbank starts from the beginning and covers most periods evenly, Spence begins with the last years of the Ming ca. 1620 and goes up to recent years. Although the title of Spence's book is evocative of the 'modernisation approach' eviscerated by Paul Cohen in the 1980s, Spence deliberately turns the issue on its head somewhat by suggesting that China in a sense has never achieved Western conceptions of 'modernity' and must be thought of mainly on its own terms. Obviously, we can still quibble a bit about what Spence does and does not emphasise – for example, Spence's account of the High Qing does not give that much consideration to its activities in Central Asia, which are a significant part of later reassessments (e.g. Peter Perdue's China Marches West (2005) or William Rowe's China's Last Empire (2009)). His periodisation of the later Qing to some extent may over-stress Sino-Western relations. Still, on the whole Spence's book is both highly readable and historiographically relevant. An added bonus is that in this particular book he often repeats himself, so as to make dipping in and out easier.

EDIT: Speaking of Rowe, if you're not absolutely committed to the 20th century, his book is an excellent introduction to the Qing Dynasty (1644-1912).