Refactor an XML to JSON parser class
It all seems really nice, I would change few things:

Firstly, your format_event_time takes a node object as a param, which is counter-intuitive as I would expect it to accept Time object. I would rename it to 'parse_xml_date` or similar and give it text instead:

def parse_xml_time(time_string)
  Time.parse(time_string).iso8601
end


The main purpose of this change is to make your build_event method the only method specifying the exact xml locations of the required properties, so it is easy to find if those changed.

Secondly,your build_event method can be slightly simplified to (including first change):

def build_event(node)
  {
    title: node.at_xpath('title').content
    url: node.at_xpath('link').content
    start: parse_xml_time(node.at_xpath('pubDate').content)
    allDay: false
  }
end


And finally I would add a small private method get_param:

def get_xml_param(attribute, node)
  node.at_xpath(attribute).content
end


Which would simplify build_event to:

def build_event(node)
  {
    title: get_xml_param('title', node)
    url: get_xml_param('url', node)
    start: parse_xml_time(get_xml_param('pubTime', node))
    allDay: false
  }
end


UPDATE:

Actually there might be few more changes. 

Your parse_events method can be written like this:

def parse_events
  xml.xpath('//item').map {|node| build_event(node) }
end


And I would consider caching an xml method:

def xml
  @xml ||= Nokogiri::XML::Document.parse(open(@url))
end