How to write an RSpec test for a simple PUT update?

You forgot to .reload your @article, and on update action your response most likely perform redirect, so

RSpec 2:

describe "PUT update/:id" do
  let(:attr) do 
    { :title => 'new title', :content => 'new content' }
  end

  before(:each) do
    put :update, :id => @article.id, :article => attr
    @article.reload
  end

  it { response.should redirect_to(@article) }
  it { @article.title.should eql attr[:title] }
  it { @article.content.should eql attr[:content] }
end

Rspec 3:

describe "PUT update/:id" do
  let(:attr) do 
    { :title => 'new title', :content => 'new content' }
  end

  before(:each) do
    put :update, :id => @article.id, :article => attr
    @article.reload
  end

  it { expect(response).to redirect_to(@article) }
  it { expect(@article.title).to eql attr[:title] }
  it { expect(@article.content).to eql attr[:content] }
end