Probably closest to an espresso is an AeroPress, which is similar to a French press, but uses manual pressure to extract more aromatic componds while being lower on bitter and sour notes than most other methods.
And of course all the "hand filter" varieties of the drip-family can be an option, if it's simply coffee, not exactly espresso you are after.
In any case you'll need a source of freshly boiled water, e.g. an electric kettle and - important for some offices - the permit to use it.
Of course a good cleaning of the office machine, adjusting it properly and filling it with good beans might be also a step in the right direction... (And have you checked - many machines do have a little hidden "door" or similar where you can use your own (ground) coffee.)