NP-hard problems on the class of caterpillars
A caterpillar is equivalent to a list of integers $n_1,n_2,...,n_m$ (given in unary), in which integer $n_i$ represents a central node $i$ and its $n_i$ neighbours.

So your question could be rephrased: "Are there hard problems in which the input is a list of (unary) integers?"

Clearly this is cheating, but a list of integers can encode everything (e.g. a binary number $(0,1,0,0,1,1,0,1)$, a string, a graph, ...), so given a polynomial time instance decoding/encoding algorithm, every problem (in particular all NP-complete problems) can be translated to a problem on a single caterpillar; note that this is not valid for problems in which the input is a single path given in unary: in this case we end up to the open (is it still open? :-) question $P=^?NP$ (through TALLY).

You could change your question to "Are there natural hard problems on caterpillars?", but like in many other contexts the word natural (probably) cannot be defined formally.

As a trivial (fairly natural :-) example of NP-hard problem: "Given a caterpillar $n_1,n_2,...,n_{3m}$ of length $3m$, can it be rearranged into a new caterpillar $n_{j_1}, n_{j_2}, ..., n_{j_{3m}}$ such that all groups of consecutive 3 central nodes (starting from the first) has the same number of total neighbours? ($n_{j_1} + n_{j_2} + n_{j_3} = n_{j_4} + n_{j_5} + n_{j_6} = ...$ and so on)