python assures me this is a valid solution:

>111211131114111511221123112411251132113311341135114211431144114511521153115411551212131214121512221223122412251232123312341235124212431244124512521253125412551313141315132213231324132513321333133413351342134313441345135213531354135514141514221423142414251432143314341435144214431444144514521453145414551515221523152415251532153315341535154215431544154515521553155415552222322242225223322342235224322442245225322542255232324232523332334233523432344234523532354235524242524332434243524432444244524532454245525253325342535254325442545255325542555333343335334433453354335534343534443445345434553535443545355435554444544554545555

625(?) presses, i was way off. :-( It's still a lot fewer than trying all 10,000 individual 4 digit possibilities that the keypad implies are there.

Also this is possibly not the shortest, according to some sibling comments to mine, above. This could be the upper bound?