Verb, transitive or intransitive. The OED’s definition:
To bring (something) up to its full capacity; to fill to the top (a partly full container, spec. (the cells of) a motor vehicle’s battery). Used esp. with reference to a drinker’s glass, freq. with the person as object.
The first citation is from a 1937 article in The Times: “In order to help the owner-driver to look after his battery, a combined acid-level indicator, vent plug and filler cup has been introduced, thus enabling the cells to be ‘topped up’ accurately and visibly, without removing the vent plugs.”
Top up is subtly different from the similar fill or fill up, indicating the the real or metaphorical receptacle is not (or not yet) empty. Replenish would probably be the closest equivalent, a word that does not trip off the tongue. Its widest use in the U.K. came from pay-as-you-go mobile phone companies, such as Virgin, who, interestingly, slightly changed the meaning. That is, there is no such thing as being “full” of minutes; topping up your mobile means simply adding more money to your account.
Chris topped up the generator with gas, spilling it on the hot metal. Then he urinated on some paint cans in the alley and locked the door. (Dan Baum, The New Yorker, September 19, 2005)/ I’m sure that the Republicans will claim savings — but those savings will come entirely from limiting the vouchers to below the rate of rise in health care costs; in effect, they will come from denying medical care to those who can’t afford to top up their premiums.(Paul Krugman, New York Times, April 4, 2011)
notoneoffbritishisms.com 
While I think the phrase works, the usual American version would be “top off.”
Another phrase that was part of my chidlhood, something that my grandfather used often. Perhaps it was because he grew up in a community which consisted largely of Irish immigrants.
In the UK, this is an incredibly common usage, both referring to mobile phones (you can ‘top up’ your prepay mobile at the hole in the wall (ATMs in US?) now), or, more usually in everyday conversation, to drinks, or other liquids. You would commonly say ‘would you like a top-up?’ when offering more wine at the dinner table for example. ‘Top off’ is used completely differently, to denote rounding off or completion, usually of a series of events. For example, it’s common to hear someone talking about their day and ending with “and to top it all off….”, almost exclusively in a negative context, similar to “the final straw”
I notice you mention that there is no “top” to “top up” to in the case of crediting a pay-as-you-go mobile phone account, which appears to be related to the example of having to top up your premiums even though that’s not actually a limit of healthcare that you can purchase. Similar examples would be to top up the payments on your endowment mortgage or pension to make up for lower than anticipated growth.