Firstly, it is important to note that, while autophagy's primary role is to 'shut down' the cell in hopes that reduced function will aid its return to homoeostasis, there is evidence of it being used as a primary method of programmed cell death in certain animals, where periods of massive cell elimination are required (e.g. insect metamorphosis). Source (Nature Review)
Your question is somewhat difficult to answer, as there is fairly limited research in the area, to my knowledge. For extrapolation purposes, it's also necessary to establish what we should consider as autophagy and apoptosis in simple organisms - how similar do the mechanisms of the processes have to be in a rudimentary form in primitive organisms to qualify as such? There is evidence of an apoptosis-like process, as well as autophagy in a range of protozoan bacteria. PLOS article (Evolution of apoptosis-like programmed cell death in unicellular protozoan parasites, Kaczanowski et. al) (Metamorphoses of Malaria: The role of autophagy in parasite differentiation, Isabelle Coppens) Additionally, based on genomic investigations, macro-/micro- autophagy are likely to be present in all non-parasitic primitive eukaryotes. (Origin and Evolution of Self-Consumption: Autophagy
Timothy Hughes and Tor Erik Rusten)
Based on some reading I did during my first degree, it is likely that both apoptosis and autophagy both developed from a precursor process that was present in prokaryotes (some bacteria, such as Myxobacteria, undergo a type of programmed cell death). An older nature article extrapolates that, based on prior research, it is likely eukaryotes came to possess the rudimentary processes of apoptosis (and likely autophagy) through the absorption of multiple strains of bacterial genomes. (Origin and evolution of eukaryotic apoptosis: the bacterial connection, E V Koonin and L Aravind)
Ultimately, I feel it's difficult to state which developed first, if, indeed, they did so separately. Assuming they originated from the same initial mechanism, which seems likely, there is a distinct possibility of them developing alongside each other. If this is the case, it's a bit of a 'chicken and egg' situation - at which point would you consider apoptosis apoptosis? There is evidence of both occurring in yeasts, as well as protozoa, so assuming these processes are indeed precursors to what we observe in multicellular eukaryotes, simultaneous development seems most likely.
Please excuse me posting the names of some articles, rather than links - I have limited use of them at the moment.