No, it's not true. It's an indisputable fact that the vast majority of the refugees who crossed the border irregularly in Hungary eventually made their way to other countries within the EU. It's true that some neighbouring countries like Austria tried to slow that down and that some also remained in the country for a period of time but it's not like Hungary was or is forced to host many Syrian refugees.
In fact, other countries tried largely unsuccessfully to force Hungary to take (some of) them back through the Dublin system, which Hungary essentially ignores. The EU Commission also had a plan to distribute refugees all over the EU, which foresaw that Hungary should take care of a mere 1.79% of whichever number of refugees would get into that system.
That plan was questionable for different reasons and there are objective factors that make it difficult for Hungary to handle a large number of refugees but you should not be mistaken about who was begging and taking care of refugees (the rest of the EU) and who was refusing all the propositions and not sharing the burden (it's Hungary).
Also the most common position in Hungary and other central European countries has never been “Give us money, we will help” or “Sorry we can only do so much” but rather “We don't agree that these people need help, they are not true refugees because they transited through Turkey, and besides we don't like Muslims so if we really have to we might take a few hundreds Christians”.