Apache 2.2 and PHP 5.3 updates in 2013 to run 3.7 properly were to be expected, its once in a blue moon.
HOWEVER, 3.8 has been a never ending nightmare. Plugins not updating, script time outs, failed install messages, over and over and over.
I can tell you with full confidence after doing this 8 times in a row on 5 different hosts (bluehost, host gator, network solutions, godaddy, and host for web)
=> go for the manual update. I think the WP team has botched something here for the first time in a decade. They are human after all.
Best advice I can give here: IGNORE ALL the crap like
define('WP_MEMORY_LIMIT', '128M'); and updating php.ini settings like max_execution_time and memory_limit... blah blah...
This is WordPress NOT Magento.
1- back up file system
2- back up database
3- do the manual update (sad but true)
I can assure you that after doing the long winded manual updates for all the plugins and the WP core that everything went well, I even went back and re-did the older installs. :)
On a side note, make sure your host is running this or newer
Apache version 2.2.23
PHP version 5.3.21
If it helps, HostGator has an INI override and NetWork solutions still NOTWORK... lol
- good luck