Your use of BesselYZero[0,1] suggests you intend to define n=0 and k=1. This choice makes u[r,\[Theta]] equal to zero since the argument of the sine is zero.

As @yarchik notes, u is a function of two variables so you can use Plot3D or ContourPlot to visualize it.

For example, with n=1, k=1 you can write the function directly from the expression you give:

Block[{R = 1, \[CapitalTheta] = 2 \[Pi], \[Alpha] = \[Pi], n = 1,k = 1},
   -2/(R \[CapitalTheta]  BesselYZero[n, k]) BesselJ[n, BesselJZero[n, k]/R  r] Sin[(n  \[Pi])/\[Alpha] \[Theta]]
]

then plot it, e.g., using N to get numerical values of the roots for faster evaluation:

Plot3D[Evaluate[N[%]], {r, 0, 3}, {\[Theta], 0, 2 \[Pi]}]

to get

Your use of BesselYZero[0,1] suggests you intend to define n=0 and k=1. This choice makes u[r,\[Theta]] equal to zero since the argument of the sine is zero.

As @yarchik notes, u is a function of two variables so you can use Plot3D or ContourPlot to visualize it.

For example, with n=1, k=1 you can write the function directly from the expression you give:

u = Block[{R = 1, \[CapitalTheta] = 2 \[Pi], \[Alpha] = \[Pi], n = 1,k = 1},
   -2/(R \[CapitalTheta]  BesselYZero[n, k]) BesselJ[n, BesselJZero[n, k]/R  r] Sin[(n  \[Pi])/\[Alpha] \[Theta]]
]

Convert to numerical values for faster plots:

uN = N[u]

then plot it, e.g.,

Plot3D[uN, {r, 0, 3}, {\[Theta], 0, 2 \[Pi]}]

to get

Or try a polar plot with

ParametricPlot3D[{r  Cos[\[Theta]], r  Sin[\[Theta]], uN}, {r, 0, 3}, {\[Theta], 0, 2 \[Pi]}, BoxRatios -> {1, 1, 1/3}]