Consider an ARMA($p,q$)-GARCH($r,s$) model \begin{aligned} r_t &= \mu_t + u_t, \\ \mu_t &= \varphi_1 r_{t-1}+\dots+\varphi_p r_{t-p} + \theta_1 u_{t-1}+\dots+\theta_q u_{t-q}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 +\dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d.(0,1). \end{aligned} Suppose you have estimated the model. The conditional mean equation is the ARMA equation for $\mu_t$. If the estimated conditional mean suits you as a point forecast (this would be the case under square loss, for example), you can ignore the other equations and only use the ARMA equation. Where the other equations play a role is the estimation stage, not the forecasting stage.

ARIMA-GARCH is mostly the same, one just has to take a cumulative sum of the corresponding first-difference series which is modelled by an ARMA-GARCH model.

Now to address your questions specifically,

When I am predicting the Yt+1 of the ARIMA model, would it be 0.62+θ0.52 or 0.23+θ0.25?

Neither of them. It would be $\hat\theta_1\cdot 0.62$.

If it is the former (i.e. 0.62+θ0.52), would that imply that the GARCH model has no effect on the first forecast of the ARIMA(0,0,1)?

If your point forecast is the conditional mean, the conditional variance equation only plays a role in the estimation stage, not the forecasting stage. But if your point forecast is something else than the conditional mean, the conditional variance equation and the distributional assumption plays a role also in the forecasting stage.

Consider an ARMA($p,q$)-GARCH($r,s$) model \begin{aligned} r_t &= \mu_t + u_t, \\ \mu_t &= \varphi_1 r_{t-1}+\dots+\varphi_p r_{t-p} + \theta_1 u_{t-1}+\dots+\theta_q u_{t-q}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 +\dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d.(0,1). \end{aligned} Suppose you have estimated the model. The conditional mean equation is the ARMA equation for $\mu_t$: $\mathbb{E}(r_t|I_{t-1})=\mu_t$ where $I_t$ is the information up to and including time $t$. If the estimated conditional mean suits you as a point forecast (this would be the case under square loss, for example), you can ignore the other equations and only use the ARMA equation. Where the other equations play a role is the estimation stage, not the forecasting stage.

ARIMA-GARCH is mostly the same, one just has to take a cumulative sum of the corresponding first-difference series which is modelled by an ARMA-GARCH model.

Now to address your questions specifically,

When I am predicting the Yt+1 of the ARIMA model, would it be 0.62+θ0.52 or 0.23+θ0.25?

Neither of them. It would be $\hat\theta_1\cdot 0.62$.

If it is the former (i.e. 0.62+θ0.52), would that imply that the GARCH model has no effect on the first forecast of the ARIMA(0,0,1)?

If your point forecast is the conditional mean, the conditional variance equation only plays a role in the estimation stage, not the forecasting stage. But if your point forecast is something else than the conditional mean, the conditional variance equation and the distributional assumption plays a role also in the forecasting stage.

Consider an ARMA($p,q$)-GARCH($r,s$) model \begin{aligned} r_t &= \mu_t + u_t, \\ \mu_t &= \varphi_1 r_{t-1}+\dots+\varphi_p r_{t-p} + \theta_1 u_{t-1}+\dots+\theta_q u_{t-q}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 +\dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d.(0,1). \end{aligned} Suppose you have estimated the model. The conditional mean equation is the ARMA equation for $\mu_t$. If the estimated conditional mean suits you as a point forecast (this would be the case under square loss, for example), you can ignore the other equations and only use the ARMA equation. (Where the other equations play a role is the estimation stage, not the forecasting stage).

ARIMA-GARCH is mostly the same, one just has to take a cumulative sum of the corresponding first-difference series which is modelled by ARMA-GARCH model.

Now to address your questions specifically,

When I am predicting the Yt+1 of the ARIMA model, would it be 0.62+θ0.52 or 0.23+θ0.25?

Neither of them. It would be $\theta_1\cdot 0.62$.

If it is the former (i.e. 0.62+θ0.52), would that imply that the GARCH model has no effect on the first forecast of the ARIMA(0,0,1)?

If your point forecast is the conditional mean, the conditional variance equation only plays a role in the estimation stage, not the forecasting stage. But if your point forecast is something else than the conditional mean, the conditional variance equation and the distributional assumption plays a role also in the forecasting stage.

Consider an ARMA($p,q$)-GARCH($r,s$) model \begin{aligned} r_t &= \mu_t + u_t, \\ \mu_t &= \varphi_1 r_{t-1}+\dots+\varphi_p r_{t-p} + \theta_1 u_{t-1}+\dots+\theta_q u_{t-q}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 +\dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d.(0,1). \end{aligned} Suppose you have estimated the model. The conditional mean equation is the ARMA equation for $\mu_t$. If the estimated conditional mean suits you as a point forecast (this would be the case under square loss, for example), you can ignore the other equations and only use the ARMA equation. Where the other equations play a role is the estimation stage, not the forecasting stage.

ARIMA-GARCH is mostly the same, one just has to take a cumulative sum of the corresponding first-difference series which is modelled by an ARMA-GARCH model.

Now to address your questions specifically,

When I am predicting the Yt+1 of the ARIMA model, would it be 0.62+θ0.52 or 0.23+θ0.25?

Neither of them. It would be $\hat\theta_1\cdot 0.62$.

If it is the former (i.e. 0.62+θ0.52), would that imply that the GARCH model has no effect on the first forecast of the ARIMA(0,0,1)?

If your point forecast is the conditional mean, the conditional variance equation only plays a role in the estimation stage, not the forecasting stage. But if your point forecast is something else than the conditional mean, the conditional variance equation and the distributional assumption plays a role also in the forecasting stage.