According to the article, Budapest could implement a bike rental scheme in "all of downtown" for the cost of a single Combino, the name of the new Siemens-built, low-floor trams that operate on the large ring road.
I checked into it myself and found that the scheme isn't exactly imminent. Balázs Tökés, bicycle affairs director at the city's transportation department, said that right now, he's waiting for bids on a call for tenders on a feasibility study. So actually, they haven't yet even studied concrete ideas for a study.
I also contacted the main source for the above-mentioned article, Ádám Bodor, deputy of bicycle affairs in the Ministry of Traffic, Communication and Energy. Bodor emailed back that he wasn't in a position to speak for the city of Budapest, but that he is doing preparatory work on EU funding (based on Structural Funds) that would be available for the project. However, the onus is on Budapest City Hall, together with affected district governments, to prepare the project and submit it for funding.
Bodor continued: "I just can tell you my personal feelings about it: I think about 4-5 million Euro would be enough to cover the city center (inside the Kis-Körút (small ring road), V. district + Jewish quarter + Buda river bank) with a rental system similiar to those in Paris and Barcelona. I would propose this system because the systems like Call-a-Bike in Germany are not secure enough for Hungarian conditions."
So Budapest's bike-hire scheme, at least in Bodor's conception, would be very modest in scale, covering only the inner core of Pest and not what I'd consider all of downtown, let's say at least everything inside the large ring road plus the Buda bank of the Danube. Granted, if it's going to happen, it has to start somewhere, but then again, if it's on too small a scale, with too few bicycle stations, it won't offer an attractive alternative to existing transport means, and it'll fade away. This is what happened in Brussels -- and now the city's trying again on a grander scale.
Interestingly, the territory envisioned by Bodor (between the small ring road and the Danube) is precisely the area under consideration by City Hall as a pedestrian-only zone. This would be a fantastic boon to traffic-clogged Budapest, in my opinion. But there is one problem: At last word, the zone would exclude bicycles, an omission that the Hungarian Cyclists Club http://www.kerekparosklub.org has been lobbying against.
My other comment on the idea is that I believe it may be putting the cart before the horse. If Budapest is going to promote bicycling as a means of transport, the most urgent task is to improve infrastructure. The paths and lanes now are very poor. At least two thirds are no more than lines on sidewalks, and in no place but on Andrássy út do they run on both sides of the street for both directions of traffic. And lastly, the paths don't form a cohesive network, so even if you were inclined to use them, you won't get far without running into a dead end. Before Budapest launches a free bike hire system, it should give people a place to ride them. Otherwise, it'll be handicapped from the get go.